Book algebra

Algebraic Geometry over the Complex Numbers

Donu Arapura

PrefaceAlgebraic geometry is the geometric study of sets of solutions to polynomial

equations over a field (or ring). These objects called algebraic varieties (or schemesor...) can be studied using tools from commutative and homological algebra. Whenthe field is the field of complex numbers, these methods can be supplemented withtranscendental ones, that is by methods from complex analysis, differential geome-try and topology. Much of the beauty of the subject stems from the rich interplayof these various techniques and viewpoints. Unfortunately, this also makes it a hardsubject to learn. This book evolved from various courses in algebraic geometry thatI taught at Purdue. In these courses, I felt my job was to act as a guide to the vastterain. I did not feel obligated to prove everything, since the standard accounts ofthe algebraic and transcendental sides of the subject by Hartshorne [Har] and Grif-fiths and Harris [GH] are remarkably complete, and perhaps a little daunting as aconsequence. In this book I have tried to maintain a reasonable balance betweenrigour, intuition and completeness; so it retains some of the informal quality oflecture notes. As for prerequisites, I have tried not to assume too much more thana mastery of standard graduate courses in algebra, analysis and topology. Conse-quently, I have included discussion of a number of topics which are technically notpart of algebraic geometry. On the other hand, since the basics are covered quickly,some prior exposure to elementary algebraic geometry and manifold theory (at thelevel of say Reid [R] and Spivak [Sk1]) would certainly be desirable.

Chapter one starts off in first gear with an extended informal introductionillustrated with concrete examples. Things only really get going in the secondchapter, where sheaves are introduced and used to define manifolds and algebraicvarieties in a unified way. A watered down notion of scheme – sufficient for ourneeds – is also presented shortly thereafter. Sheaf cohomology is developed quicklyfrom scratch in chapter 4, and applied to De Rham theory and Riemann surfacesin the next few chapters. By part three, we have hit highway speeds with Hodgetheory. This is where algebraic geometry meets differential geometry on the onehand, and some serious homological algebra on the other. Although I have skirtedaround some of the analysis, I did not want to treat this entirely as a black box. Ihave included a sketch of the heat equation proof of the Hodge theorem, which Ithink is reasonably accessible and quite pretty. This theorem along with the hardLefschetz theorem has some remarkable consequences for the geometry and topologyof algebraic varietes. I discuss some of these applications in the remaining chapters.The starred sections can be skipped without losing continuity. In the fourth part, Iwant to consider some methods for actually computing Hodge numbers for variousexamples, such as hypersurfaces. A primarily algebraic approach is used here. Thisis justified by the GAGA theorem, whose proof is outlined. The books ends atthe shores of some really deep water, with an explanation of some conjectures ofGrothendieck and Hodge along with a context to put them in.

I would like to thank Bill Butske, Ed Dunne, Harold Donnelly, Anton Fonarev,Su-Jeong Kang, Mohan Ramachandran, Peter Scheiblechner, Darren Tapp, andRazvan Veliche for catching some errors. My thanks also to the NSF for theirsupport over the years.

For updates check http://www.math.purdue.edu/∼dvb/book.html

http://www.math.purdue.edu/~dvb/book.html

Contents

Part 1. A Quick Tour 1

Chapter 1. Plane Curves 31.1. Conics 31.2. Singularities 41.3. Bezout’s theorem 61.4. Cubics 71.5. Quartics 91.6. Hyperelliptic curves 11

Part 2. Sheaves and Geometry 15

Chapter 2. Manifolds and Varieties via Sheaves 172.1. Sheaves of functions 172.2. Manifolds 192.3. Algebraic varieties 222.4. Stalks and tangent spaces 262.5. Singular points 292.6. Vector fields and bundles 302.7. Compact complex manifolds and varieties 33

Chapter 3. Basic Sheaf Theory 373.1. The Category of Sheaves 373.2. Exact Sequences 403.3. Direct and Inverse images 423.4. The notion of a scheme 433.5. Gluing schemes 473.6. Sheaves of Modules 493.7. Differentials 53

Chapter 4. Sheaf Cohomology 574.1. Flasque Sheaves 574.2. Cohomology 594.3. Soft sheaves 624.4. C∞-modules are soft 644.5. Mayer-Vietoris sequence 65

Chapter 5. De Rham cohomology of Manifolds 695.1. Acyclic Resolutions 695.2. De Rham’s theorem 715.3. Poincare duality 73

3

4 CONTENTS

5.4. Gysin maps 765.5. Fundamental class 785.6. Lefschetz trace formula 80

Chapter 6. Riemann Surfaces 836.1. Topological Classification 836.2. Examples 856.3. The ∂-Poincare lemma 886.4. ∂-cohomology 896.5. Projective embeddings 916.6. Function Fields and Automorphisms 946.7. Modular forms and curves 96

Chapter 7. Simplicial Methods 1017.1. Simplicial and Singular Cohomology 1017.2. H∗(Pn,Z) 1057.3. Cech cohomology 1067.4. Cech versus sheaf cohomology 1097.5. First Chern class 111

Part 3. Hodge Theory 115

Chapter 8. The Hodge theorem for Riemannian manifolds 1178.1. Hodge theory on a simplicial complex 1178.2. Harmonic forms 1188.3. The Heat Equation* 121

Chapter 9. Toward Hodge theory for Complex Manifolds 1279.1. Riemann Surfaces Revisited 1279.2. Dolbeault’s theorem 1289.3. Complex Tori 130

Chapter 10. Kahler manifolds 13310.1. Kahler metrics 13310.2. The Hodge decomposition 13510.3. Picard groups 137

Chapter 11. Homological methods in Hodge theory 13911.1. Pure Hodge structures 13911.2. Canonical Hodge Decomposition 14011.3. Hodge decomposition for Moishezon manifolds 14411.4. Hypercohomology* 14611.5. Holomorphic de Rham complex* 14911.6. The Deligne-Hodge decomposition* 150

Chapter 12. A little algebraic surface theory 15512.1. Examples 15512.2. The Neron-Severi group 15812.3. Adjunction and Riemann-Roch 16012.4. The Hodge index theorem 16212.5. Fibred surfaces 163

CONTENTS 5

Chapter 13. Topology of families 16513.1. Fibre bundles 16513.2. Monodromy of families of elliptic curves 16513.3. Local systems 16813.4. Higher direct images* 17013.5. First Betti number of a fibred variety* 172

Chapter 14. The Hard Lefschetz Theorem 17514.1. Hard Lefschetz 17514.2. Proof of Hard Lefschetz 17614.3. Barth’s theorem* 17814.4. Lefschetz pencils* 17914.5. Degeneration of Leray* 18314.6. Positivity of higher Chern classes* 185

Part 4. Coherent Cohomology 189

Chapter 15. Coherent sheaves on Projective Space 19115.1. Cohomology of line bundles on Pn 19115.2. Coherence 19315.3. Coherent Sheaves on Pn 19515.4. Cohomology of coherent sheaves 19815.5. GAGA 200

Chapter 16. Computation of some Hodge numbers 20516.1. Hodge numbers of Pn 20516.2. Hodge numbers of a hypersurface 20816.3. Hodge numbers of a hypersurface II 20916.4. Double covers 21116.5. Griffiths Residues 212

Chapter 17. Deformation invariance of Hodge numbers 21517.1. Families of varieties via schemes 21517.2. Semicontinuity of coherent cohomology 21717.3. Proof of the semicontinuity theorem 21917.4. Deformation invariance of Hodge numbers 221

Part 5. A Glimpse Beyond* 223

Chapter 18. Analogies and Conjectures 22518.1. Counting points and Euler characteristics 22518.2. The Weil conjectures 22718.3. A transcendental analogue of Weil’s conjecture 22918.4. Conjectures of Grothendieck and Hodge 23018.5. Problem of Computability 233

Bibliography 235

Index 239

Part 1

A Quick Tour

CHAPTER 1

Plane Curves

Algebraic geometry is geometry. This sounds like a tautology, but it is easy toforget when one is learning about sheaves, cohomology, Hodge structures and soon. So perhaps it is a good idea to keep ourselves grounded by taking a very quicktour of algebraic curves in the plane. We use fairly elementary and somewhat adhoc methods for now. Once we have laid the proper foundations in later chapters,we will revisit these topics and supply some of the missing details.

1.1. Conics

A complex affine plane curve is the set of zeros

X = V (f) = (x, y) ∈ C2 | f(x, y) = 0of a nonconstant polynomial f(x, y) ∈ C[x, y]. Notice that we call this a curve sinceit has one complex dimension. However, we will be slightly inconsistent and refer tothis occasionally as a surface. X is called a conic if f is a quadratic polynomial. Thestudy of conics is something one learns in school, but usually over R. The complexcase is actually easier. The methods one learns there can be used to classify conicsup to an affine transformation:(

xy

)7→(a11x+ a12y + b1a21x+ a22y + b2

)with det(aij) 6= 0. There are three possibilities

1. A union of two lines.2. A circle x2 + y2 = 13. A parabola y = x2.

The first case can of course be subdivided into the subcases of two parallel lines,two lines which meet, or a single line.

Things become simpler if we add a line at infinity. This can be achieved bypassing to the projective plane CP2 = P2 which is the set of lines in C3 containingthe origin. To any (x, y, z) ∈ C3 − 0, there corresponds a unique point [x, y, z] =span(x, y, z) ∈ P2. We embed C2 ⊂ P2 as an open set by (x, y) 7→ [x, y, 1]. Theline at infinity is given by z = 0. The closure of an affine plane curve X = V (f) inP2 is the projective plane curve

X = [x, y, z] ∈ P2 | F (x, y, z) = 0where

F (x, y, z) = zdeg ff(x/z, y/z)is the homogenization.

The game is now to classify the projective conics up to a projective lineartransformation. The projective linear group PGL3(C) = GL3(C)/C∗ acts on P2

3

4 1. PLANE CURVES

via the standard GL3(C) action on C3. The list simplifies to three cases including allthe degenerate cases: a single line, two distinct lines which meet, or the projectivizedparabola C given by

x2 − yz = 0

If we allow nonlinear transformations, then things simply further. The map fromthe projective line to the plane given by [s, t] 7→ [st, s2, t2] gives a bijection of P1 toC. The inverse can be expressed as

[x, y, z] 7→

[y, x] if (y, x) 6= 0[x, z] if (x, z) 6= 0

Note that these expressions are consistent by the equation. These formulas showthat C is homeomorphic, and in fact isomorphic in a sense to be explained in thenext chapter, to P1. Topologically, this is just the two-sphere S2.

Exercises

1. Show that the subgroup of PGL3(C) fixing the line at infinity is the groupof affine transformations.

2. Deduce the classification of projective conics from the classification of qua-dratic forms over C.

3. Deduce the classification of affine conics from exercise 1.

1.2. Singularities

We recall a version of the implicit function theorem

Theorem 1.2.1. If f(x, y) is a polynomial such that fy(0, 0) = ∂f∂y (0, 0) 6= 0,

then in a neighbourhood of (0, 0), V (f) is given by the graph of an analytic functiony = φ(x).

In outline, we can use Newton’s method. Set φ0(x) = 0, and

φn+1(x) = φn(x)− f(x, φn(x))fy(x, φn(x))

Then φn will converge to φ. Proving this requires some care of course.A point (a, b) on an affine curve X = V (f) is a singular point if

∂f

∂x(a, b) =

∂f

∂y(a, b) = 0

otherwise it is nonsingular. In a neighbourhood of a nonsingular point, we can usethe implicit function theorem to write x or y as an analytic function of the othervariable. So locally at such a point, X looks like a disk. By contrast the nodalcurve y2 = x2(x + 1) looks like a union of two disks in a neighbourhood of (0, 0)(figure 1).

The two disks are called branches of the singularity. Singularities may have onlyone branch, as in the case of the cusp y2 = x3 (figure 2).

1.2. SINGULARITIES 5

Figure 1. Nodal Curve

Figure 2. Cuspidal Curve

These pictures (figure 1, 2) are over the reals, but the singularities are complex.We can get a better sense of this, by intersecting the singularity f(x, y) = 0 witha small 3-sphere |x|2 + |y|2 = ε2 to get a circle S1 embedded in S3 in the caseof one branch. The circle is unknotted, when this is nonsingular. But it would beknotted in general [Mr]. For the cusp, we would get a so called trefoil knot (figure3).

X is called nonsingular if all its points are. The projective curve X is nonsingu-lar if of all of its points including points at infinity are nonsingular. Nonsingularityat these points can be checked, for example, by applying the previous definitionto the affine curves F (1, y, z) = 0 and F (x, 1, y) = 0. A nonsingular curve is anexample of a Riemann surface or a one dimensional complex manifold.

Exercises

1. Prove the convergence of Newton’s method in the ring of formal powerseries C[[x]], where φn → 0 if and only if the degree of its leading term

6 1. PLANE CURVES

Figure 3. Trefoil

→∞. Note that this ring is equipped with the x-adic topology, where theideals (xN ) form a fundamental system of neighbourhoods of 0.

2. Prove that Fermat’s curve xn + yn + zn = 0 in P2 is nonsingular.

1.3. Bezout’s theorem

An important feature of the projective plane is that any two lines meet. Infact, it has a much stronger property:

Theorem 1.3.1 (Weak Bezout’s theorem). Any two curves in P2 intersect.

We give an elementary classical proof here using resultants. This works overany algebraically closed field. Given two polynomials

f(y) = yn + an−1yn−1 + . . .+ a0 =

n∏i=1

(y − ri)

g(y) = ym + bm−1ym−1 + . . .+ b0 =

m∏j=1

(y − sj)

Their resultant is the expression

Res(f, g) =∏ij

(ri − sj)

It is obvious that Res(f, g) = 0 if and only if f and g have a common root. Fromthe way we have written it, it is also clear that Res(f, g) is a polynomial of degreemn in r1, . . . rn, s1, . . . sm which is symmetric separately in the r’s and s’s. So itcan be rewritten as a polynomial in the elementary symmetric polynomials in ther’s and s’s. In other words, Res(f, g) is a polynomial in the coefficients ai and bj .Standard formulas for it can be found for example in [Ln].

Proof of theorem. After translating the line at infinity if necessary, we canassume that the affine curves are given by f(x, y) = 0 and g(x, y) = 0 with bothpolynomials nonconstant in x and y. Treating these as polynomials in y withcoefficients in C[x], the resultant Res(f, g)(x) can be regarded as a nonconstantpolynomial in x. As C is algebraically closed, Res(f, g)(x) must have a root, saya. Then f(a, y) = 0 and g(a, y) = 0 have a common solution.

1.4. CUBICS 7

Suppose that the curves C,D ⊂ P2 are irreducible and distinct, then it is notdifficult to see that C ∩ D is finite. So we can ask how many points are in theintersection. To get a more refined answer, we can assign a multiplicity to thepoints of intersection. If the curves are defined by polynomials f(x, y) and g(x, y)with a common isolated zero at the origin O = (0, 0). Then define the intersectionmultiplicity at O by

iO(C,D) = dim C[[x, y]]/(f, g)where C[[x, y]] is the ring of formal power series in x and y. The ring of conver-gent power series can be used instead, and it would lead to the same result. Themultiplicities can be defined at other points by a similar procedure. While this def-inition is elegant, it does not give us much geometric insight. Here is an alternative:ip(D,E) is the number of points close to p in the intersection of small perturbationsof these curves. More precisely,

Lemma 1.3.2. ip(D,E) is the number of points in f(x, y) = ε ∩ g(x, y) =η ∩Bδ(p) for small |ε|, |η|, δ, where Bδ(p) is a δ-ball around p.

Proof. This follows from [F1, 1.2.5e].

There is another nice interpretation of this number worth mentioning. IfK1,K2 ⊂ S3 are disjoint knots, perhaps with several components, their linkingnumber is roughly the number of times one of them passes through the other. Aprecise definition can be found in any book on knot theory.

Theorem 1.3.3. Given a small sphere S3 about p, ip(D,E) is the linking num-ber of D ∩ S3 and E ∩ S3,

Proof. See [F1, 19.2.4].

We can now state the strong form of Bezout’s theorem (cor. 12.2.5).

Theorem 1.3.4 (Bezout’s theorem). The sum of intersection multiplicities atpoints of C ∩D equals the product of degrees degC · degD.

(The degrees of C and D are simply the degrees of their defining polynomials.)

Corollary 1.3.5. #C ∩D ≤ degC · degD

Exercises

1. Show that the vector space C[[x, y]]/(f, g) considered above is finite dimen-sional if f = 0 and g = 0 have an isolated zero at (0, 0).

2. Suppose that f = y. Using the original definition show that iO(C,D)equals the multiplicity of the root x = 0 of g(x, 0). Now prove Bezout’stheorem when C is a line.

1.4. Cubics

We now turn our attention to the very rich subject of cubic curves. Theseare also called elliptic curves because of their relation to elliptic functions andintegrals. In the degenerate case if the polynomial factors into a product of a linearand quadratic polynomial, then the curve is a union of a line with a conic. So nowassume that f is irreducible, then

8 1. PLANE CURVES

Lemma 1.4.1. After a projective linear transformation an irreducible cubic canbe transformed into the projective closure of an affine curve of the form y2 = p(x),where p(x) is a cubic polynomial. This is nonsingular if and only if p(x) has nomultiple roots.

Proof. See [Si1, III §1].

We note that nonsingular cubics are very different from conics, even topologi-cally.

Proposition 1.4.2. A nonsingular cubic X is homeomorphic to a torus S1 ×S1.

There is a standard way to visualize this. Take two copies of P1 correspondingto the sheets of the covering y2 = p(x). Mark four points a, b, c, d on each copycorresponding to the roots of p(x) together with ∞. Join these points pairwise byarcs, slit them, and join the two copies to get a torus.

d

a

b c

Figure 4. Visualizing the cubic

We will not attempt to make this description rigorous. Instead, we use a pa-rameterization by elliptic functions. By applying a further projective linear trans-formation, we can put our equation for X in Weierstrass form

(1.4.1) y2 = 4x3 − a2x− a3

with discriminant a32 − 27a2

3 6= 0. The idea is to parameterize the cubic by theelliptic integral ∫ z

z0

dx

y=∫ z

z0

dx√4x3 − a2x− a3

However, these integrals are not well defined since they depend on the path ofintegration. We can get around the problem by working by modulo the set of

1.5. QUARTICS 9

periods L ⊂ C, which is the set of integrals of dx/y around closed loops on X. Theset L is actually a subgroup. This follows from the fact that the set of equivalenceclasses of loops on X can be given the structure of an abelian group H1(X,Z) andthat the map γ 7→

∫γdx/y gives a homomorphism of H1(X,Z) → C. L can be

characterized in a different way.

Theorem 1.4.3. There exists a unique lattice L ⊂ C, i.e. an Abelian subgroupgenerated by two R-linearly independent numbers, such that

a2 = g2(L) = 60∑

λ∈L,λ6=0

λ−4

a4 = g3(L) = 140∑

λ∈L,λ6=0

λ−6

Proof. [Si2, I 4.3].

Fix the period lattice L as above. The Weierstrass ℘-function is given by

℘(z) =1z2

+∑

λ∈L, λ 6=0

(1

(z − λ)2− 1λ2

)This converges to an elliptic function which means that it is meromorphic on Cand doubly periodic: ℘(z + λ) = ℘(z) for λ ∈ L [Si1]. This function satisfies theWeierstrass differential equation

(℘′)2 = 4℘3 − g2(L)℘2 − g3(L)

Thus ℘ gives exactly the parameterization we wanted. We get an embedding ofC/L→ P2 given by

z 7→

[℘(z), ℘′(z), 1] if z /∈ L[0, 1, 0] otherwise

The image is the cubic curve X defined by (1.4.1). This shows that X is a torustopologically as well as analytically. See [Si1, Si2] for further details.

Exercises

1. Prove that the projective curve defined by y2 = p(x) is nonsingular if andonly if p(x) has no repeated roots.

2. Prove the singular projective curve y2z = x3 is homeomorphic to thesphere.

1.5. Quartics

We now turn to the degree 4 case. We claim that a nonsingular quartic inP2 is a three holed or genus 3 surface. A heuristic argument is as follows. Letf ∈ C[x, y, z] be the defining equation of our nonsingular quartic, and let g =(x3 + y3 + z3)x. The degenerate quartic g = 0 is the union of a nonsingular cubicand a line. Topologically this is a torus meeting a sphere in 3 points (figure 5).Consider the pencil tf + (1 − t)g. As t evolves from t = 0 to 1, the 3 points ofintersection open up into circles, resulting in a genus 3 surface (figure 6).

10 1. PLANE CURVES

Figure 5. Degenerate Quartic

Figure 6. Nonsingular Quartic

In going from degree 3 to 4, we seemed to have skipped over genus 2. It is possibleto realize such a surface in the plane, but only by allowing singularities. Considerthe curve X ⊂ P2 given by

x2z2 − y2z2 + x2y2 = 0

This has a single singularity at the origin [0, 0, 1]. To analyze this, switch to affinecoordinates by setting z = 1. Then polynomial x2 − y2 + x2y2 is irreducible, so itcannot be factored into polynomials, but it can be factored into convergent powerseries

x2 − y2 + x2y2 = (x+ y +∑

aijxiyj)︸︷︷︸

f

(x− y +∑

bijxiyj)︸︷︷︸

g

By the implicit function theorem, the branches f = g = 0 are local analyticallyequivalent to disks. It follows that in a neighbourhood of the origin, the curve lookslike two disks touching at a point. We get a genus 2 surface by pulling these apart(figure 7).

Figure 7. Normalization of Singular Quartic

The procedure of pulling apart the points described above can be carried outwithin algebraic geometry. It is called normalization:

Theorem 1.5.1. Given a curve X, there exists a nonsingular curve X and aproper surjective morphism H : X → X which is finite to one everywhere, and one

1.6. HYPERELLIPTIC CURVES 11

to one over all but finitely many points. This is uniquely characterized by theseproperties.

The word “morphism” will not be defined precisely until the next chapter. Forthe present, we should understand it to be a map definable by algebraic expressionssuch as polynomials. We sketch the construction. Further details will be given lateron. Given an integral domain R with field of fractions K, the integral closure of Ris the set of elements x ∈ K such that xn + an−1x

n−1 + . . . a0 = 0 for some ai ∈ R.This is closed under addition and multiplication, therefore it forms a ring [AM,chap 5]. The basic facts can be summarized as follows:

Theorem 1.5.2. If f ∈ C[x, y] is an irreducible polynomial. Then the integralclosure R of the domain R = C[x, y]/(f) is finitely generated as an algebra. IfC[x1, . . . xn]→ R is a surjection, and f1, . . . fN generators for the kernel. Then

V (f1, . . . fN ) = (a1, . . . an) | fi(a1, . . . an) = 0

is nonsingular in the sense that the Jacobian has expected rank (§2.5).

Proof. See [AM, prop 9.2] and [E, cor. 13.13]

Suppose that X = V (f). Then X = V (f1, . . . fN ). We can lift the inclusionR ⊂ R to a homomorphism of polynomial rings by completing the diagram

C[x1, . . . xn] // R

C[x, y] //

h

OO

R

OO

This determines a pair of polynomials h(x), h(y) ∈ C[x1 . . . xn], which gives a poly-nomial map H : Cn → C2. By restriction, we get our desired map H : X → X.This is the construction in the affine case. In general, we proceed by gluing theseaffine normalizations together. The precise construction will be given in §3.5.

Exercises

1. Verify that x2z2−y2z2+x2y2 = 0 is irreducible and has exactly one singularpoint.

2. Verify that x2 − y2 + x2y2 can be factored as above using formal powerseries.

3. Show that t = y/x lies in the integral closure R of C[x, y]/(y2 − x3). Showthat R ∼= C[t].

4. Show that t = y/x lies in the integral closure R of C[x, y]/(x2− y2 +x2y2).Show that R ∼= C[x, t]/(1− t2 − x2t2).

1.6. Hyperelliptic curves

An affine hyperelliptic curve is a curve of the form y2 = p(x) where p(x) hasdistinct roots. The associated hyperelliptic curve X is gotten by taking the closurein P2 and then normalizing to obtain a nonsingular curve. Once again we start bydescribing the topology. It would be a genus g surface, that is a surface with gholes, where

12 1. PLANE CURVES

Proposition 1.6.1. The genus g of X is given by ddeg p(x)/2e − 1, where d emeans round up to the nearest integer.

We can see this by using a cut and paste construction generalizing what we didfor cubics. Let a1, . . . an denote the roots of p(x) if deg p(x) is even, or the rootstogether with ∞ otherwise. Take two copies of P1 and make slits from a1 to a2,a3 to a4 and so on, and then join them along the slits. The genus of the result isn/2− 1.

Corollary 1.6.2. Every natural number is the genus of some algebraic curve.

The term hyperelliptic is generally reserved for g > 1 or equivalently deg p(x) >4. The original interest in hyperelliptic curves stemmed from the study of integralsof the form

∫q(x)dx/

√p(x). As with cubics, these are only well defined modulo

their periods Lq = ∫γq(x)dx/

√p(x) | γ closed. However, this is usually no longer

a discrete subgroup, so C/Lq would be a very strange object. What turns out tobe better is to consider all these integrals simultaneously.

Theorem 1.6.3. The set

L =

(∫

γ

xidx√p(x)

)0≤i<g

| γ closed

is a lattice in Cg.

This will follow almost immediately from the Hodge decomposition (theo-rem 10.2.4). We thus get a well defined map from X to the torus J(X) = Cg/Lgiven by

α(x) =

(∫ x

x0

xidx√p(x)

)mod L

J(X) is called the Jacobian of X, and α is called the Abel-Jacobi map. Togetherthese form one of the cornerstones of algebraic curve theory.

We can make this more explicit in an example. Consider the curve X definedby y2 = x6 − 1. This has genus two, so that J(X) is a two dimensional torus. LetE be the elliptic curve given by v2 = u3 − 1. We have two morphisms πi : X → Edefined by

π1 : u = x2, v = y

π2 : u = x−2, v =√−1yx−3

The second map appears to have singularities, but one can appeal to either generaltheory or explicit calculation to show that it is defined everywhere. We can seethat the differential du/v on E pulls back to 2xdx/y and 2

√−1dx/y under π1 and

π2 respectively. Combining these yields a map π1 × π2 : X → E × E under whichthe lattice defining E × E corresponds to a sublattice L′ ⊆ L. Therefore

J(X) = C2/L = (C2/L′)/(L/L′) = E × E/(L/L′)

We express this relation by saying that J(X) is isogenous to E×E, which means thatit is a quotient of the second by a finite abelian group. The relation is symmetric,but this is not obvious. It is worthwhile understanding what is going on a moreabstract level. The lattice L can be identified with either the first homology group

1.6. HYPERELLIPTIC CURVES 13

H1(X,Z) or the first cohomology group H1(X,Z) by Poincare duality. Using thesecond description, we have a map

H1(E,Z)⊕H1(E,Z) ∼= H1(E × E,Z)→ H1(X,Z)

and L′ is the image.

Exercises The curve X above can be constructed explicitly by gluing charts definedby y2 = x6 − 1 and y2

2 = 1− x62 via x2 = x−1, y2 = yx−3

1. Check that X is nonsingular and that it maps onto the projective closureof y2 = x6 − 1.

2. Using these coordinates, show that the above formulas for πi define mapsfrom X to the projective closure of v2 = u3 − 1.

Part 2

Sheaves and Geometry

CHAPTER 2

Manifolds and Varieties via Sheaves

In rough terms, a manifold is a “space” which looks locally like Euclidean space.An algebraic variety can be defined similarly as a “space” which looks locally likethe zero set of a collection of polynomials. Point set topology alone would not besufficient to capture this notion of space. These examples come with distinguishedclasses of functions (C∞ functions in the first case, and polynomials in the second),and we want these classes to be preserved under the above local identifications.Sheaf theory provides a natural language in which to make these ideas precise. Asheaf on a topological space X is essentially a distinguished class of functions, orthings which behave like functions, on open subsets of X. The main requirement isthat the condition to be distinguished is local, which means that it can be checkedin a neighbourhood of every point of X. For a sheaf of rings, we have an additionalrequirement, that the distinguished functions on U ⊆ X should form a commutativering. With these definitions, the somewhat vague idea of a space can replaced bythe precise notion of a concrete ringed space, which consists of topological spacetogether with a sheaf of rings of functions. Both manifolds and varieties are concreteringed spaces.

Sheaves were first defined by Leray in the late 1940’s. They played a key role indevelopment of algebraic and complex analytic geometry, in the pioneering works ofCartan, Grothendieck, Kodaira, Serre, Spencer, and others in the following decade.Although it is rarely presented this way in introductory texts (e.g. [Sk2, Wa]).basic manifold theory can also be developed quite naturally in this framework. Inthis chapter we want to lay the basic foundation for rest of the book. The goal hereis to introduce the language of sheaves, and then to carry out a uniform treatmentof real and complex manifolds and algebraic varieties from this point of view. Thisapproach allows us to highlight the similarities, as well as the differences, amongthese spaces.

2.1. Sheaves of functions

As we said in the introduction, we need to define sheaves in order to eventuallydefine manifolds and varieties. We start with a more primitive notion. In manyparts of mathematics, we encounter topological spaces with distinguished classes offunctions on them: continuous functions on topological spaces, C∞-functions on Rn,holomorphic functions on Cn and so on. These functions may have singularities, sothey may only be defined over subsets of the space; we will primarily be interestedin the case where these subsets are open. We say that such a collection of functionsis a presheaf if it is closed under restriction. Given sets X and T , let MapT (X)denote the set of maps from X to T . Here is the precise definition of a presheaf.

17

18 2. MANIFOLDS AND VARIETIES VIA SHEAVES

Definition 2.1.1. Suppose that X is a topological space and T a set. A presheafof T -valued functions on X is a collection of subsets P(U) ⊆ MapT (U), for eachnonempty open U ⊆ X, such that the restriction f |V ∈ P(V ) whenever f ∈ P(U)and V ⊂ U .

If the defining conditions for P(U) are local, which means that they can bechecked in a neighbourhood of a point, then P is called a sheaf. More precisely:

Definition 2.1.2. A presheaf of functions P is called a sheaf if given any openset U with an open cover Ui, a function f on U lies in P(U) if f |Ui ∈ P(Ui) forall i.

Example 2.1.3. Let X be a topological space and T a set with at least twoelements t1, t2. Let T pre(U) be the set of constant functions from U to T . This iscertainly a presheaf, since the restriction of constant function is constant. However,if X contains a disconnected open set U , then we can write U = U1∪U2 as a unionof two disjoint open sets. The function τ taking the value of ti on Ui is not inT pre(U), but τ |Ui

∈ T pre(Ui). Therefore T pre is not sheaf.

However, there is a simple remedy.

Example 2.1.4. A function is locally constant if it is constant in a neighbour-hood of a point. For instance, the function τ constructed above is locally constantbut not constant. The set of locally constant functions, denoted by T (U) or TX(U),is a now sheaf, precisely because the condition can be checked locally. A sheaf ofthis form is called a constant sheaf.

Example 2.1.5. Let T be a topological space, then the set of continuous func-tions ContX,T (U) from U ⊆ X to T is a sheaf, since continuity is a local condition.When T is discrete, continuous functions are the same thing as locally constantfunctions, so we recover the previous example.

Example 2.1.6. Let X = Rn, the sets C∞(U) of C∞ real valued functionsform a sheaf.

Example 2.1.7. Let X = Cn, the sets O(U) of holomorphic functions on Uform a sheaf. (A function of several variables is holomorphic if it is C∞ andholomorphic in each variable.)

Example 2.1.8. Let L be a linear differential operator on Rn with C∞ coeffi-cients (e. g.

∑∂2/∂x2

i ). Let S(U) denote the space of C∞ solutions in U . This isa sheaf.

Example 2.1.9. Let X = Rn, the sets L1(U) of L1 or summable functionsforms a presheaf which is not a sheaf. The point is that summability is a globalcondition and not a local one.

We can always create a sheaf from a presheaf by the following construction.

Example 2.1.10. Given a presheaf P of functions from X to T . Define the

Ps(U) = f : U → T | ∀x ∈ U,∃ a neighbourhood Ux of x, such that f |Ux∈ P(Ux)

This is a sheaf called the sheafification of P.

2.2. MANIFOLDS 19

When P is a presheaf of constant functions, Ps is exactly the sheaf of locallyconstant functions. When this construction is applied to the presheaf L1, we obtainthe sheaf of locally L1 functions.

Exercises

1. Check that Ps is a sheaf.2. Let π : B → X be a surjective continuous map of topological spaces. Prove

that the presheaf of sections

B(U) = σ : U → B | σ continuous, ∀x ∈ U, π σ(x) = xis a sheaf.

3. Let F : X → Y be surjective continuous map. Suppose that P is a sheafof T -valued functions on X. Define f ∈ Q(U) ⊂ MapT (U) if and only ifits pullback F ∗f = f F |f−1U ∈ P(F−1(U)). Show that Q is a sheaf on Y .

4. Let Y ⊂ X be a closed subset of a topological space. Let P be a sheaf ofT -valued functions on X. For each open U ⊂ Y , let PY (U) be the set offunctions f : U → T locally extendible to an element of P, i.e. f ∈ PY (U)if and only there for each y ∈ U , there exists a neighbourhood V ⊂ X andan element of P(V ) restricting to f |V ∩U . Show that PY is a sheaf.

2.2. Manifolds

Let k be a field. Then Mapk(X) is a commutative k-algebra with pointwiseaddition and multplication.

Definition 2.2.1. Let R be a sheaf of k-valued functions on X. We say thatR is a sheaf of algebras if each R(U) ⊆Mapk(U) is a subalgebra. We call the pair(X,R) a concrete ringed space over k, or simply a k-space.

(Rn, ContRn,R), (Rn, C∞) are examples of R-spaces, and (Cn,O) is an exampleof a C-space.

Definition 2.2.2. A morphism of k-spaces (X,R) → (Y,S) is a continuousmap F : X → Y such that f ∈ S(U) implies F ∗f ∈ R(F−1U), where F ∗f =f F |f1U .

For example, a C∞ map F : Rn → Rm induces a morphism (Rn, C∞) →(Rm, C∞) of R-spaces, and a holomorphic map F : Cn → Cm induces a morphismof C-spaces. The converse is also true, and will be left for the exercises.

This is good place to introduce, or perhaps remind the reader of, the notion of acategory. A category C consists of a set (or class) of objects ObjC and for each pairA,B ∈ C, a set HomC(A,B) of morphisms from A to B. There is a compositionlaw

: HomC(B,C)×HomC(A,B)→ HomC(A,C),and distinguished elements idA ∈ HomC(A,A) which satisfy

(C1) associativity: f (g h) = (f g) h,(C2) identity: f idA = f and idA g = g,

whenever these are defined. Categories abound in mathematics. A basic exampleis the category of Sets. The objects are sets, HomSets(A,B) is just the set of mapsfrom A to B, and composition and idA have the usual meanings. Similarly, wecan form the category of groups and group homomorphisms, the category of rings


and rings homomorphisms, and the category of topological spaces and continuousmaps. We have essentially constructed another example. We can take the class ofobjects to be k-spaces, and morphisms as above. These can be seen to constitute acategory, once we observe that the identity is a morphism and the composition ofmorphisms is a morphism.

The notion of an isomorphism makes sense in any category. We will spell thisout for k-spaces.

Definition 2.2.3. An isomorphism of k-spaces (X,R) ∼= (Y,S) is a homeo-morphism F : X → Y such that f ∈ S(U) if and only if F ∗f ∈ R(F−1U).

Given a sheaf S on X and open set U ⊂ X, let S|U denote the sheaf on Udefined by V 7→ S(V ) for each V ⊆ U .

Definition 2.2.4. An n-dimensional C∞ manifold is an R-space (X,C∞X ) suchthat

1. The topology of X is given by a metric1.2. X admits an open cover Ui such that each (Ui, C∞X |Ui

) is isomorphic to(Bi, C∞|Bi

) for some open balls Bi ⊂ Rn.

The isomorphisms (Ui, C∞|Ui) ∼= (Bi, C∞|Bi

) correspond to coordinate chartsin more conventional treatments. The collection of all such charts is called anatlas. Given a coordinate chart, we can pull back the standard coordinates fromthe ball to Ui. So we always have the option of writing expressions locally in thesecoordinates.

There are a number of variations on this idea:

Definition 2.2.5.1. An n-dimensional topological manifold is defined as above but with (Rn, C∞)

replaced by (Rn, ContRn,R).2. An n-dimensional complex manifold can be defined by replacing (Rn, C∞)

by (Cn,O).

The one-dimensional complex manifolds are usually called Riemann surfaces.

Definition 2.2.6. A C∞ map from one C∞ manifold to another is just amorphism of R-spaces. A holomorphic map between complex manifolds is definedas a morphism of C-spaces.

The class of C∞ manifolds and maps form a category; an isomorphism in thiscategory is called a diffeomorphism. Likewise, the class of complex manifolds andholomorphic maps forms a category, with isomorphisms called biholomorphisms. Bydefinition any point of a manifold has neigbourhood, called a coordinate neigbour-hood, diffeomorphic or biholomorphic to a ball. Given a complex manifold (X,OX),we say that f : X → R is C∞ if and only if f g is C∞ for each holomorphic mapg : B → X from a coordinate ball B ⊂ Cn. We state for the record:

Lemma 2.2.7. An n-dimensional complex manifold together with its sheaf ofC∞ functions is a 2n-dimensional C∞ manifold.

1 It is equivalent and perhaps more standard to require that the topology is Hausdorff and

paracompact. (The paracompactness of metric spaces is a theorem of A. Stone [Ke]. In theopposite direction a metric is given by the Riemannian distance associated to a Riemannian

metric constructed using a partition of unity.)

2.2. MANIFOLDS 21

Proof. An n-dimensional complex manifold (X,OX) is locally biholomorphicto a ball in Cn, and hence (X,C∞X ) is locally diffeomorphic to the same ball regardedas a subset of R2n.

Later on, we will need to write things in coordinates. The pull back of thestandard coordinates on a ball B ⊂ Cn under local biholomorphism from X ⊃B′ ∼= B, are referred to as local analytic coordinates on X. We typically denotethis by z1, . . . zn. Then the real and imaginary parts x1 = Re(z1), y1 = Im(z1), . . .give local coordinates for the underlying C∞-manifold.

Let us consider some examples of manifolds. Certainly any open subset of Rn(or Cn) is a (complex) manifold in an obvious fashion. To get less trivial examples,we need one more definition.

Definition 2.2.8. Given an n-dimensional C∞ manifold X, a closed subsetY ⊂ X is called a closed m-dimensional submanifold if for any point x ∈ Y, thereexists a neighbourhood U of x in X and a diffeomorphism to a ball B ⊂ Rn con-taining 0, such that Y ∩ U maps to the intersection of B with an m-dimensionallinear subspace. A similar definition holds for complex manifolds.

When we use the word “submanifold” without qualification, we will alwaysmean “closed submanifold”. Given a closed submanifold Y ⊂ X, define C∞Y tobe the sheaf of functions which are locally extendible to C∞ functions on X. Fora complex submanifold Y ⊂ X, we define OY to be the sheaf of functions whichlocally extend to holomorphic functions.

Lemma 2.2.9. If Y ⊂ X is a closed submanifold of a C∞ (respectively complex)manifold, then (Y,C∞Y ) (respectively (Y,OY )) is also a C∞ (respectively complex)manifold.

Proof. We treat the C∞ case, the holomorphic case is similar. Choose a localdiffeomorphism (X,C∞X ) to a ball B ⊂ Rn such that Y correponds to a B ∩ Rm.Then any C∞ function f(x1, . . . xm) on B ∩ Rm extends trivially to a C∞ on Band conversely. Thus (Y,C∞Y ) is locally diffeomorphic to a ball in Rm.

With this lemma in hand, it is possible to produce many interesting examplesof manifolds starting from Rn. For example, the unit sphere Sn−1 ⊂ Rn, whichis the set of solutions to

∑x2i = 1, is an (n − 1)-dimensional manifold. Further

examples are given in the exercises.The following example is of fundamental importance in algebraic geometry.

Example 2.2.10. Complex projective space PnC = CPn is the set of one dimen-sional subspaces of Cn+1. (We will usually drop the C and simply write Pn unlessthere is danger of confusion.) Let π : Cn+1 − 0 → Pn be the natural projectionwhich sends a vector to its span. In the sequel, we usually denote π(x0, . . . xn) by[x0, . . . xn]. Pn is given the quotient topology which is defined so that U ⊂ Pn isopen if and only if π−1U is open. Define a function f : U → C to be holomorphicexactly when f π is holomorphic. Then the presheaf of holomorphic functions OPn

is a sheaf, and the pair (Pn,OPn) is a complex manifold. In fact, if we set

Ui = [x0, . . . xn] | xi 6= 0,then the map

[x0, . . . xn] 7→ (x0/xi, . . . xi/xi . . . xn/xi)induces an isomorphism Ui ∼= Cn The notation . . . x . . . means skip x in the list.


Exercises

1. Let T = Rn/Zn be a torus. Let π : Rn → T be the natural projection. ForU ⊆ T , define f ∈ C∞T (U) if and only if the pullback f π is C∞ in theusual sense. Show that (T,C∞T ) is a C∞ manifold.

2. Let τ be a nonreal complex number. Let E = C/(Z + Zτ) and π : C→ Edenote the projection. Define f ∈ OE(U) if and only if the pullback f πis holomorphic. Show that E is a Riemann surface. Such a surface is calledan elliptic curve.

3. Show that a map F : Rn → Rm is C∞ in the usual sense if and only if itinduces a morphism (Rn, C∞)→ (Rm, C∞) of R-spaces.

4. Assuming the implicit function theorem [Sk1, p 41], check that f−1(0) isa closed n− 1 dimensional submanifold of Rn provided that f : Rn → R isC∞ function such that the gradient (∂f/∂xi) does not vanish along f−1(0).In particular, show that the quadric defined by x2

1+x22+. . .+x2

k−x2k+1 . . .−

x2n = 1 is a closed n− 1 dimensional submanifold of Rn for k ≥ 1.

5. Let f1, . . . fr be C∞ functions on Rn, and let X be the set of commonzeros of these functions. Suppose that the rank of the Jacobian (∂fi/∂xj)is n − m at every point of X. Then show that X is an m dimensionalsubmanifold using the implicit function theorem.

6. Apply the previous exercise to show that the set O(n) of orthogonal ma-trices n× n matrices is a submanifold of Rn2

. Show that the matrix mul-tiplication and inversion are C∞ maps. A manifold which is also a groupwith C∞ group operations is called a Lie group. So O(n) is an example.

7. The complex Grassmanian G = G(2, n) is the set of 2 dimensional sub-spaces of Cn. Let M ⊂ C2n be the open set of 2 × n matrices of rank 2.Let π : M → G be the surjective map which sends a matrix to the spanof its rows. Give G the quotient topology induced from M , and definef ∈ OG(U) if and only if π f ∈ OM (π−1U). For i 6= j, let Uij ⊂M be theset of matrices with (1, 0)t and (0, 1)t for the ith and jth columns. Showthat

C2n−4 ∼= Uij ∼= π(Uij)

and conclude that G is a 2n− 4 dimensional complex manifold.

2.3. Algebraic varieties

Standard references for this material are [EH], [Hrs], [Har], [Mu2] and [Sh].Let k be a field. Affine space of dimension n over k is defined as Ank = kn. Whenk = R or C, we can endow this space with the standard topology induced by theEuclidean metric, and we will refer to this as the classical topology. At the otherextreme is the Zariski topology which makes sense for any k. On A1

k = k, the opensets consists of complements of finite sets together with the empty set. In general,this topology can be defined to be the weakest topology for which the polynomialsAnk → k are continuous functions. The closed sets of Ank are precisely the sets ofzeros

V (S) = a ∈ An | f(a) = 0∀f ∈ Sof sets of polynomials S ⊂ R = k[x1, . . . xn]. Sets of this form are also calledalgebraic. The Zariski topology has a basis given by open sets of the form D(g) =

2.3. ALGEBRAIC VARIETIES 23

X − V (g), g ∈ R. If U ⊆ Ank is open, a function F : U → k is called regular if itcan be expressed as a ratio of polynomials F (x) = f(x)/g(x), such that g has nozeros on U .

Lemma 2.3.1. Let OAn(U) denote the set of regular functions on U . ThenU 7→ OAn(U) is a sheaf of k-algebras. Thus (Ank ,OAn) is a k-space.

Proof. It is clearly a presheaf. Suppose that F : U → k is represented by aratio of polynomials fi/gi on Ui ⊆ U . Since k[x1, . . . , xn] has unique factorization,we can assume that these are reduced fractions. Since fi(x)/gi(x) = fj(x)/gj(x)for all x ∈ Ui ∩ Uj . Equality holds as elements of k(x1, . . . , xn). Therefore, we canassume that fi = fj and gi = gj . Thus F ∈ OX(U).

The ringed space (An,OAn) is the basic object, which plays the role of (Rn, C∞)or (Cn,O) in manifold theory. However, unlike the case of manifolds, algebraic va-rieties are more complicated than Ank , even locally. The local models are given byaffine algebraic varieties. These are irreducible algebraic subsets of some Ank . Recallthat irreducibility means that the set cannot be written as a union of two proper al-gebraic sets. At this point it will convenient to assume that k is algebraically closed,and keep this assumption for the remainder of this section. Hilbert’s nullstellensatz[AM, E] shows that the map I 7→ V (I) gives a one to one correspondence betweenvarious algebraic and geometric objects. We summarize all this in the table below:

Algebra Geometrymaximal ideals of R points of An

prime ideals in R irreducible algebraic subsets of Anradical ideals in R algebraic subsets of An

Fix an affine algebraic variety X ⊆ Ank . We give X the induced topology. This isalso called the Zariski topology. Given an open set U ⊂ X, a function F : U → kis regular if it is locally extendible to a regular function on an open set of An asa defined above. That is, if every point of U has an open neighbourhood V ⊂ Ankwith a regular function G : V → k, for which F = G|V ∩U .

Lemma 2.3.2. Let X be an affine variety, and let OX(U) denote the set ofregular functions on U . Then U → OX(U) is a sheaf of k-algebras. If I is theprime ideal for which X = V (I), then OX(X) ∼= k[x0, . . . , xn]/I.

Proof. The sheaf property ofOX is clear from § 2.1 exercise 4 and the previouslemma. So it is enough to prove the last statement. Let R = k[x0, . . . , xn]/I.Clearly there is an injection of R → OX(X) given by sending a polynomial to thecorresponding regular function. Suppose that F ∈ OX(X). Let J = g ∈ R | gF ∈R. This is an ideal, and it suffices to show that 1 ∈ J . By assumption, for anya ∈ X there exists polynomials f, g such that g(a) 6= 0 and F (x) = f(x)/g(x) forall x in a neighbourhood of a. Therefore g ∈ J , where g is the image of g in R.But this implies that V (J) = ∅, and so by the nullstellensatz 1 ∈ J .

Thus an affine variety gives rise to a k-space (X,OX). The ring of global regularfunctions O(X) = OX(X) is an integral domain called the coordinate ring of X.Its field of fractions k(X) is called the function field of X, and it can be identifiedwith the field of rational functions on X. The coordinate ring determines (X,OX)completely as we shall see later.


In analogy with manifolds, we can define an (abstract) algebraic variety as ak-space which is locally isomorphic to an affine variety, and which satisfies someversion of the Hausdorff condition. It will be convenient to break the definition intoparts.

Definition 2.3.3. A prevariety over k is a k-space (X,OX) such that X isconnected and there exists a finite open cover Ui such that each (Ui,OX |Ui

) isisomorphic, as a k-space, to an affine variety. A morphism of prevarieties is amorphism of the underlying k-spaces.

Before going further, let us consider the most important non-affine example.

Example 2.3.4. Let Pnk be the set of one-dimensional subspaces of kn+1. Usingthe natural projection π : An+1−0 → Pn, give Pn the quotient topology ( U ⊂ Pnis open if and only if π−1U is open). Equivalently, the closed sets of Pn are zerosof sets of homogeneous polynomials in k[x0, . . . xn]. Define a function f : U → k tobe regular exactly when f π is regular. Such a function can be represented as theratio

f π(x0, . . . xn) =p(x0, . . . , xn)q(x0, . . . , xn)

of two homogeneous polynomials of the same degree such that q has no zeros onπ−1U . Then the presheaf of regular functions OPn is a sheaf, and the pair (Pn,OPn)is easily seen to be a prevariety with affine open cover Ui as in example 2.2.10.

We now have to explain the “Hausdorff” or separation axiom. In fact, theZariski topology is never actually Hausdorff (except in trivial cases), so we have tomodify the definition without losing the essential meaning. First observe that theusual Hausdorff condition for a space X is equivalent to the requirement that thediagonal ∆ = (x, x) |x ∈ X is closed in X × X with its product topology. Inthe case of prevarieties, we have to be careful about what we mean by products.We expect An × Am = An+m, but should notice that the topology on this spaceis not the product topology. The safest way to define products is in terms of auniversal property. The collection of prevarieties and morphisms forms a category.The following can be found in [Mu2, I§6]:

Proposition 2.3.5. Let (X,OX) and (Y,OY ) be prevarieties. Then the Carte-sian product X × Y carries a topology and a sheaf of functions OX×Y such that(X × Y,OX×Y ) is a prevariety and the projections to X and Y are morphisms. If(Z,OZ) is any prevariety which maps via morphisms f and g to X and Y then themap f × g : Z → X × Y is a morphism.

Thus (X × Y,OX×Y ) is the product in the category theoretic sense (see [Ln,chap 1] for an explanation of this). If X ⊂ An and Y ⊂ Am are affine, then there isno mystery as to what the product means, it is simply the Cartesian product X×Ywith the structure of a variety coming from the embedding X × Y ⊂ An+m. Theproduct Pn×Pm can be also be constructed explicitly by using the Segre embeddingPn × Pm ⊂ P(n+1)(m+1)−1 given by

([x0, . . . , xn], [y0, . . . , ym]) 7→ [x0y0, x0y1, . . . , xnym]

Definition 2.3.6. A prevariety X is an variety, or algebraic variety, if thediagonal ∆ ⊂ X ×X is closed. A map between varieties is called a morphism ora regular map if it is a morphism of prevarieties.

2.3. ALGEBRAIC VARIETIES 25

Affine spaces are varieties in Serre’s sense, since the diagonal ∆ ⊂ A2nk =

Ank × Ank is the closed set defined by xi = xi+n. Projective spaces also satisfythe seperation condition for similar reasons. Further examples can be producedby taking suitable subsets. Let (X,OX) be an algebraic variety over k. A closedirreducible subset Y ⊂ X is called a closed subvariety. Given an open set U ⊂Y define OY (U) to be the set functions which are locally extendible to regularfunctions on X.

Proposition 2.3.7. Suppose that Y ⊂ X is a closed subvariety of an algebraicvariety. Then (Y,OY ) is an algebraic variety.

Proof. Let Ui be an open cover of X by affine varieties. Choose an em-bedding Ui ⊂ ANk as a closed subset. Then Y ∩ Ui ⊂ ANk is also embedded as aclosed set, and the restriction OY |Y ∩Ui

is the sheaf of functions on Y ∩ Ui whichare locally extendible to regular functions on ANk . Thus (Y ∩ Ui,OY |Y ∩Ui

) is anaffine variety. This implies that Y is a prevariety. Denoting the diagonal of X andY by ∆X and ∆Y respectively. We see that ∆Y = ∆X ∩Y ×Y is closed in X ×X,and therefore in Y × Y .

It is worth making the description of projective varieties, or closed subvarietiesof projective space, more explicit. A nonempty subset of An+1

k is conical if itcontains 0 and is stable the action of λ ∈ k∗ given by v 7→ λv. Given X ⊂ Pnk ,CX = π−1X ∪ 0 ⊂ An+1

k is conical, and all conical sets arise this way. If I ⊆S = k[x0, . . . xn] is a homogeneous ideal, then V (I) is conical and so correspondsto a closed subset of Pnk . From the nullstellensatz, we obtain a dictionary similarto the earlier one. (The maximal ideal of the origin S+ = (x0, . . . , xn) needs to beexcluded in order to get a bijection.)

Algebra Geometryhomogeneous radical ideals in S other than S+ algebraic subsets of Pnhomogeneous prime ideals in S other than S+ algebraic subvarieties of Pn

Given a subvariety X ⊆ Pnk , the elements of OX(U) are functions f : U → k suchthat f π is regular. Such a function can be represented locally as the ratio of twohomogeneous polynomials of the same degree.

When k = C, we can use the stronger classical topology on PnC introduced in2.2.10. This is inherited by subvarieties, and is also called the classical topology.When there is danger of confusion, we write Xan to indicate, a variety X with itsclassical topology. (The superscript an stands for “analytic”.)

Exercises

1. Let X be an affine variety with coordinate ring R and function field K.Show that X is homeomorphic to MspecR, which is the set of maximalideals of R with closed sets given by V (I) = m | m ⊃ I for ideals I ⊂ R.Given m ∈ MspecR, define Rm = g/f | f, g ∈ R, f /∈ m. Show thatOX(U) is isomorphic to ∩m∈URm.

2. Give a complete proof that Pn is an algebraic variety.3. Given an open subset U of an algebraic variety X. Let OU = OX |U . Prove

that (U,OU ) is a variety. An open subvariety of a projective variety iscalled quasi-projective.


4. Make the Grassmanian Gk(2, n), which is the set of 2 dimensional subspacesof kn, into a prevariety by imitating the constructions of the exercises inSection 2.2.

5. Check that Gk(2, n) is a variety.6. After identifying k6 ∼= ∧2k4, Gk(2, 4) can be embedded in P5

k, by sendingthe span of v, w ∈ k4 to the line spanned by ω = v∧w. Check that this is amorphism and that the image is a subvariety given by the Plucker equationω∧ω = 0. Write this out as a homogeneous quadratic polynomial equationin the coordinates of ω.

7. An algebraic group is an algebraic geometer’s version of a Lie group. Itis a variety G which is also a group such that the group operations aremorphisms. Show that G = GLn(k) is an algebraic group.

8. Given an algebraic group G, an action on a variety X is a morphism G×X → X denoted by “·” such that (gh) · x = g · (h · x). A variety is calledhomogeneous if an algebraic group acts transitively on it. Check that affinespaces, projective spaces, and Grassmanians are homogeneous.

2.4. Stalks and tangent spaces

Given two functions defined in possibly different neighbourhoods of a pointx ∈ X, we say they have the same germ at x if their restrictions to some commonneighbourhood agree. This is is an equivalence relation. The germ at x of a functionf defined near X is the equivalence class containing f . We denote this by fx.

Definition 2.4.1. Given a presheaf of functions P, its stalk Px at x is the setof germs of functions contained in some P(U) with x ∈ U .

It will be useful to give a more abstract characterization of the stalk usingdirect limits (which are also called inductive limits, or filtered colimits). We explaindirect limits in the present context, and refer to [E, appendix 6] or [Ln] for a morecomplete discussion. Suppose that a set L is equipped with a family of mapsP(U) → L, where U ranges over open neighbourhoods of x. We will say thatthe family is a compatible family if P(U) → L factors through P(V ), wheneverV ⊂ U . The maps P(U) → Px given by f 7→ fx forms a compatible family. Aset L equipped with a compatible family of maps is called a direct limit of P(U) ifand only if for any M with a compatible family P(U)→M , there is a unique mapL→M making the obvious diagrams commute. This property characterizes L upto isomorphism, so we may speak of the direct limit

lim−→x∈UP(U).

Lemma 2.4.2. Px = lim−→x∈U P(U).

Proof. Suppose that φ : P(U) → M is a compatible family. Then φ(f) =φ(f |V ) whenever f ∈ P(U) and x ∈ V ⊂ U . Therefore φ(f) depends only on thegerm fx. Thus φ induces a map Px →M as required.

When R is a sheaf of algebras of functions, then Rx is an algebra. In mostof the examples considered earlier, Rx is a local ring i.e. it has a unique maximalideal. This follows from:

2.4. STALKS AND TANGENT SPACES 27

Lemma 2.4.3. Rx is a local ring if and only if the following property holds: Iff ∈ R(U) with f(x) 6= 0, then 1/f |V is defined and lies in R(V ) for some open setx ∈ V ⊆ U .

Proof. Let mx be the set of germs of functions vanishing at x. For Rx to belocal with maximal ideal mx, it is necessary and sufficient that each f ∈ Rx −mx

is invertible. The last condition is the equivalent to the existence of an inverse1/f |V ∈ R(V ) for some V .

Definition 2.4.4. We will say that a k-space is locally ringed if each of thestalks are local rings.

C∞manifolds, complex manifolds and algebraic varieties are all locally ringedby the above lemma. When (X,OX) is an n-dimensional complex manifold, thelocal ring OX,x can be identified with ring of convergent power series in n variables.When X is a variety, the local ring OX,x is also well understood. We may replaceX by an affine variety with coordinate ring R = OX(X). Consider the maximalideal

mx = f ∈ R | f(x) = 0then

Lemma 2.4.5. OX,x is isomorphic to the localization

Rmx =g

f| f, g ∈ R, f /∈ mx

.

Proof. Let K be the field of fractions of R. A germ in OX,x is representedby a regular function defined in a neighbourhood of x, but this is fraction f/g ∈ Kwith g /∈ mx.

By standard commutative algebra [AM, cor. 7.4], the stalks of algebraic vari-eties are Noetherian since they are localizations of Noetherian rings. This is alsotrue for complex manifolds, although the argument is bit more delicate [GPR, p.12]. By contrast, when X is a C∞-manifold, the stalks are non-Noetherian localrings. This is easy to check by a theorem of Krull [AM, pp 110-111] which im-plies that a Noetherian local ring R with maximal ideal m satisfies ∩nmn = 0.However, if R is the ring of germs of C∞ functions, then the intersection ∩nmn

contains nonzero functions such ase−1/x2

if x > 00 otherwise

Nevertheless, the maximal ideals are finitely generated.

Proposition 2.4.6. If R is the ring of germs at 0 of C∞ functions on Rn,then its maximal ideal m is generated by the coordinate functions x1, . . . xn.

Proof. See exercises.

If R is a local ring with maximal ideal m then k = R/m is a field called theresidue field. We will often convey all this by referring to the triple (R,m, k) asa local ring. In order to talk about tangent spaces in this generallity, it will beconvenient to introduce the following axioms:


Definition 2.4.7. We will say a local ring (R,m, k) satisfies the tangent spaceconditions if

1. There is an inclusion k ⊂ R which gives a splitting of the natural mapR→ k.

2. The ideal m is finitely generated.

For stalks of C∞ and complex manifolds and algebraic varieties over k, theresidue fields are respectively, R,C and k. The inclusion of germs of constantfunctions gives the first condition in these examples, and the second was discussedabove.

Definition 2.4.8. When (R,m, k) is a local ring satisfying the tangent spaceconditions, we define its cotangent space as T ∗R = m/m2 = m⊗R k, and its tangentspace as TR = Hom(T ∗R, k). When X is a manifold or variety, we write Tx = TX,x(respectively T ∗x = T ∗X,x) for TOX,x

(respectively T ∗OX,x).

When (R,m, k) satisfies the tangent space conditions, R/m2 splits canonicallyas k ⊕ T ∗x .

Definition 2.4.9. With R as above, given f ∈ R, define its differential df asthe projection of (f mod m2) to T ∗x under the above decomposition.

To see why this terminology is justified, we compute the differential when R isthe ring of germs of C∞ functions on Rn at 0. Then f ∈ R can be expanded usingTaylor’s formula

f(x1, . . . , xn) = f(0) +∑ ∂f

∂xi

∣∣∣∣0

xi + r(x1, . . . , xn)

where the remainder r lies in m2. Therefore df coincides with the image of thesecond term on the right, which is the usual expression

df =∑ ∂f

∂xi

∣∣∣∣0

dxi

Lemma 2.4.10. d : R→ T ∗R is a k-linear derivation, i.e. it satisfies the Leibnizrule d(fg) = f(x)dg + g(x)df .

As a corollary, it follows that a tangent vector v ∈ TR = T ∗∗R gives rise to aderivation δv = v d : R→ k.

Lemma 2.4.11. The map v 7→ δv yields an isomorphism between TR and thevector space Derk(R, k) of k-linear derivations from R to k.

Proof. Given δ ∈ Derk(R, k), we can see that δ(m2) ⊆ m. Therefore itinduces a map v : m/m2 → R/m = k, and we can check that δ = δv.

Lemma 2.4.12. When (R,m, k) is the ring of germs at 0 of C∞ functions onRn (or holomorphic functions on Cn, or regular functions on Ank). Then a basisfor Derk(R, k) is given

Di =∂

∂xi

∣∣∣∣0

i = 1, . . . n

Exercises

2.5. SINGULAR POINTS 29

1. Prove proposition 2.4.6. (Hint: given f ∈ m, let

fi =∫ 1

0

∂f

∂xi(tx1, . . . txn) dt

show that f =∑fixi.)

2. Prove Lemma 2.4.10.3. Let F : (X,R)→ (Y, S) be a morphism of k-spaces. If x ∈ X and y = F (x),

check that the homomorphism F ∗ : Sy → Rx taking a germ of f to thegerm of f F is well defined. When X and S are both locally ringed, showthat F ∗ is local, i.e. F ∗(my) ⊆ mx where m denotes the maximal ideals.

4. When F : X → Y is a C∞ map of manifolds, use the previous exerciseto construct the induced linear map dF : Tx → Ty. Calculate this for(X,x) = (Rn, 0) and (Y, y) = (Rm, 0) and show that this is given by amatrix of partial derivatives.

5. Check that with the appropriate identification given a C∞ function on Xviewed as a C∞ map from f : X → R, df in the sense of 2.4.9 and in thesense of the previous exercise coincide.

6. Check that the operation (X,x) 7→ Tx determines a functor on the categoryof C∞-manifolds and base point preserving maps. (The definition of functorcan be found in §3.1.) Interpret this as the chain rule.

7. Given a Lie group G with identity e. An element g ∈ G acts on G byh 7→ ghg−1. Let Ad(g) : Te → Te be the differential of this map. Showthat Ad defines a homomorphism from G to GL(Te) called the adjointrepresentation.

2.5. Singular points

Algebraic varieties can be very complicated, even locally. We want to say thata point of a variety over an algebraically closed field k is nonsingular or smooth ifit looks like affine space at a microscopic level. Otherwise, the point is singular.When k = C, a variety should be a manifold in a neighbourhood of a nonsingularpoint. The implicit function suggests a way to make this condition more precise.Suppose that X ⊆ ANk is a closed subvariety defined by the ideal (f1, . . . , fr) andlet x ∈ X. Then x ∈ X should be nonsingular if the Jacobian matrix ( ∂fi

∂xj|x) has

the expected rank N − dimX, where dimX can be defined as the transcendencedegree of the function field k(X) over k. We can reformulate this in a more intrinsicfashion thanks to the following:

Lemma 2.5.1. The vector space TX,x is isomorphic to the kernel of ( ∂fi

∂xj|x).

Proof. Let R = OAN ,x, S = OX,x ∼= R/(f1, . . . fr) and π : R → S be thenatural map. We also set J = ( ∂fi

∂xj|x). Then any element δ′ ∈ Derk(S, k) gives

a derivation δ′ π ∈ Derk(R, k), which vanishes only if δ′ vanishes. A derivationin δ ∈ Derk(R, k) comes from S if and only if δ(fi) = 0 for all i. We can use thebasis ∂/∂xj |x to identify Derk(R, k) with kN . Putting all of this together gives a


commutative diagram

0 // Der(S, k) // Der(R, k)δ 7→δ(fi)//

∼=

kr

=

kN

J // kr

from which the lemma follows.

Definition 2.5.2. A point x on a (not necessarily affine) variety is called anonsingular or smooth point if and only dimTX,x = dimX, otherwise x is calledsingular. X is nonsingular or smooth, if every point is nonsingular.

The condition for nonsingularity of x is usually formulated as saying the localring OX,x is a regular local ring [AM, E]. But this is equivalent to what wasgiven above, since dimX coincides with the Krull dimension [loc. cit.] of the ringOX,x. Affine and projective spaces and Grassmanians are examples of nonsingularvarieties.

Over C, we can use the holomorphic implicit function theorem to obtain:

Proposition 2.5.3. Given a subvariety X ⊂ CN and a point x ∈ X. Thepoint x is nonsingular if and only if there exists a neighbourhood U of x in CN forthe classical topology, such that X ∩U is a closed complex submanifold of CN , withdimension equal to dimX.

Proof. See [Mu2, III.4] for details.

Corollary 2.5.4. Given a nonsingular algebraic subvariety X of AnC or PnC,the space Xan is a complex submanifold of Cn or PnC .

Exercises

1. If f(x0, . . . xn) is a homogeneous polynomial of degree d, prove Euler’sformula

∑xi

∂f∂xi

= d · f(x0, . . . xn). Use this to show that the point p onthe projective hypersurface defined by f is singular if and only if all thepartials of f vanish at p. Determine the set of singular points defined byx5

0 + . . .+ x54 − 5x0 . . . x4 in P4

C.2. Prove that if X ⊂ ANk is a variety then dimTx ≤ N . Give an example of a

curve in ANk for which equality is attained, for N = 2, 3, . . ..3. Let f(x0, . . . , xn) = 0 define a nonsingular hypersurface X ⊂ PnC. Show

that there exist a hyperplane H such that X ∩H is nonsingular. This is aspecial case of Bertini’s theorem.

4. Show that a homogeneous variety nonsingular.

2.6. Vector fields and bundles

A C∞ vector field on a manifold X is a choice vx ∈ Tx, for each x ∈ X, whichvaries in a C∞ fashion. The dual notion is that of a 1-form (or covector field). Let〈, 〉 denote the pairing between Tx and T ∗x . We can make these notions precise asfollows:

2.6. VECTOR FIELDS AND BUNDLES 31

Definition 2.6.1. A C∞ vector field on X is a collection of vectors vx ∈ Tx,x ∈ X, such that the map x 7→ 〈vx, dfx〉 lies in C∞(U) for each open U ⊆ X andf ∈ C∞(U). A C∞ 1-form is a collection ωx ∈ T ∗x such that x 7→ 〈vx, ωx〉 ∈ C∞(X)for every C∞-vector field.,

Given a C∞-function f on X, we can define df = x 7→ dfx. The definition isrigged up to ensure df is an example of a C∞ 1-form, and that locally any 1-formis a linear combination of forms of this type. More precisely, if U is a coordinateneighbourhood with coordinates x1, . . . xn, then any 1-form on U can be expandedas∑fi(x1, . . . xn)dxi with C∞-coefficients. Similarly, vector fields on U are given

by sums∑fi∂/∂xi. Let T (X) and E1(X) denote the space of C∞-vector fields and

1-forms on X. These are modules over the ring C∞(X) and we have an isomorphismE1(X) ∼= HomC∞(X)(T (X), C∞(X)). The maps U 7→ T (U) and U 7→ E1(U) areeasily seen to be sheaves (of respectively ∪Tx and ∪T ∗x valued functions) on Xdenoted by TX and E1

X respectively.There is another standard approach to defining vector fields on a manifold X.

The disjoint union of the tangent spaces TX =∐x Tx can be assembled into a

manifold called the tangent bundle TX , which comes with a projection π : TX → Xsuch that Tx = π−1(x). We define the manifold structure on TX in such a waythat the vector fields correspond to C∞ cross sections. The tangent bundle is anexample of a structure called a vector bundle. In order to give the general definitionsimultaneously in several different categories, we will fix a choice of:

(a) a C∞-manifold X and the standard C∞-manifold structure on k = R,(b) a C∞-manifold X and the standard C∞-manifold structure on k = C,(c) a complex manifold X and the standard complex manifold structure on

k = C,(d) or an algebraic variety X with an identification k ∼= A1

k.A rank n vector bundle is a map π : V → X which is locally a product X×kn → X.Here is the precise definition.

Definition 2.6.2. A rank n vector bundle on X is a morphism π : V → Xsuch that there exists an open cover Ui of X and commutative diagrams

π−1Ui

""EEEEEEEE φi

∼= // Ui × kn

wwwwwwwww

Ui

such that the isomorphisms

φi φ−1j : Ui ∩ Uj × kn ∼= Ui ∩ Uj × kn

are k-linear on each fibre. A bundle is called C∞ real in case (a), C∞ complex incase (b), holomorphic in case (c), and algebraic in case (d). A rank 1 vector bundlewill also be called a line bundle.

The product X×kn is an example of a vector bundle called the trivial bundle ofrank n. A simple nontrivial example to keep in mind is the Mobius strip which is areal line bundle over the circle. The datum (Ui, φi) is called a local trivialization.Given a C∞ real vector bundle π : V → X, define the presheaf of sections

V(U) = s : U → π−1U | s is C∞, π s = idU


This is easily seen to be a sheaf. When V = X × Rn is the trivial vector bundle,then V(U) is the space of vector valued C∞-functions on U . In general, a sections ∈ V(U) is determined by the collection of vector-valued functions on Ui∩U givenby projecting φi s to Rn. Thus V(U) has a natural R-vector space structure.Later on, we will characterize the sheaves, called locally free sheaves, that arisefrom vector bundles in this way.

Theorem 2.6.3. Given an n-dimensional manifold X there exists a C∞ realvector bundle TX of rank n, called the tangent bundle, whose sheaf of sections isexactly TX .

Proof. Complete details for the construction of TX can be found in [Sk2,Wa]. We outline a construction when X ⊂ RN is a submanifold.2 Fix standardcoordinates y1, . . . , yN on RN . We define TX ⊂ X × RN such that (p; v1, . . . vN ) ∈TX if and only ∑

vi∂f

∂yj

∣∣∣∣p

= 0

whenever f is a C∞-function defined in a neighbourhood of p in RN such thatX ∩ U ⊆ f−1(0). We have the obvious projection π : TX → X. A sum v =∑j gj∂/∂yj , with gj ∈ C∞(U), defines a vector field on U ⊆ X precisely when

(p; g1(p), . . . gN (p)) ∈ TX for a p ∈ U . In other words, vector fields are sections ofTX .

It remains to find a local trivialization. We can find an open cover Ui of Xby coordinate neighbourhoods. Choose local coordinates x(i)

1 , . . . , x(i)n in each Ui.

Then the map

(p;w) 7→

p; ∂yj

∂x(i)k

∣∣∣∣∣p

w

identifies Ui × Rn with π−1Ui.

Tangent bundles also exist for complex manifolds, and nonsingular algebraicvarieties. However, we will postpone the construction. An example of an algebraicvector bundle of fundamental importance is the tautological line bundle L. Projec-tive space Pnk is the set of lines ` in kn+1 through 0, and we can choose each lineas a fibre of L. That is

L = (x, `) ∈ kn+1 × Pnk |x ∈ `

Let π : L → Pnk be given by projection onto the second factor. Then L is a rankone algebraic vector bundle, or line bundle, over Pnk . When k = C this can alsobe regarded as holomorphic line bundle or a C∞ complex line bundle. L is oftencalled the universal line bundle for the following reason:

Theorem 2.6.4. If X is a compact C∞ manifold with a C∞ complex linebundle p : M → X. Then for n 0, there exists a C∞ map f : X → PnC, called aclassifying map, such that M is isomorphic (as a bundle) to the pullback

f∗L = (v, x) ∈ L×X |π(v) = f(x) → X

2According to Whitney’s embedding theorem, every manifold embeds into a Euclidean space.So, in fact, this is no restriction at all.

2.7. COMPACT COMPLEX MANIFOLDS AND VARIETIES 33

Proof. Here we consider the dual line bundle M∗. Sections of this correspondto C-valued functions on M which are linear on the fibres. By compactness, we canfind finitely many sections f0, . . . , fn ∈M∗(X) which do not simultaneously vanishat any point x ∈ X. Thus we get an injective bundle map M → X × Cn given byv 7→ (f0(v), . . . , fn(v)). Under a local trivialization M |Ui

∼→ Ui×C, we can identifythe fj with C-valued functions X. Therefore we can identify M |Ui

with the spanof (f0(x), . . . , fn(x)) in Ui × Cn.

The mapsx 7→ [f0(x), . . . , fn(x)] ∈ Pn, x ∈ Ui

are independent of the choice of trivialization. So this gives a map from f : X → Pn.The pullback f∗L can also be described as the sub line bundle of X ×Cn spannedby (f0(x), . . . , fn(x)). So this coincides with M .

Exercises

1. Show that v =∑fi(x) ∂

∂xiis a C∞ vector field on Rn in the sense of

Definition 2.6.1 if and only if the coefficients fi are C∞.2. Check that L is an algebraic line bundle.3. Given a vector bundle π : V → X over a manifold, and C∞ map f : Y → X.

Show that the set

f∗V = (y, v) ∈ Y × V |π(v) = f(y)with its first projection to Y determines a vector bundle.

4. Let G = G(2, n) be the Grassmanian of 2 dimensional subspaces of kn. Thisis an algebraic variety by the exercises in section 2.3. Let S = (x, V ) ∈kn × G | x ∈ V . Show that the projection S → G is an algebraic vectorbundle of rank 2. This is called the universal bundle of rank 2 on G.

5. Prove the analogue of theorem 2.6.4 for rank 2 vector bundles: Any ranktwo C∞ complex vector bundle on a compact C∞ manifold X is isomorphicto the pullback of the universal bundle for some C∞ map X → G(2, n) withn 0.

2.7. Compact complex manifolds and varieties

Up to this point, we have been treating C∞ and complex manifolds in parallel.However, there are big differences, owing to the fact that holomorphic functions aremuch more rigid than C∞ functions. We illustrate this in a couple of ways. In par-ticular, we will see that the (most obvious) holomorphic analogue to Theorem 2.6.4would fail. We start by proving some basic facts about holomorphic functions inmany variables.

Theorem 2.7.1. Let ∆n be an open polydisk, that is a product of disks, in Cn.(1) If two holomorphic functions on ∆n agree on a nonempty open set, then

they agree on all of ∆n.(2) The maximum principle: If the absolute value of a holomorphic function f

on ∆n attains a maximum in ∆n, then f is constant on ∆n.

Proof. This can be reduced to the corresponding statements in one variable[Al]. We leave the first statement as an exercise. For the second, suppose that |f |attains a maximum at (a1, . . . an) ∈ ∆. The maximum principle in one variable


implies that f(z, a2, . . . an) is constant. Fixing z1 ∈ ∆, we see that f(z1, z, a3 . . .)is constant, and so on.

We will see in the exercises that all C∞-manifolds carry nonconstant globalC∞-functions. By contrast:

Proposition 2.7.2. If X is a compact connected complex manifold, then allholomorphic functions on X are constant.

Proof. Let f : X → C be holomorphic. Since X is compact, |f | attains amaximum at, say, x0 ∈ X. Therefore f constant in a neighbourhood of x0 by themaximum principle. Given x ∈ X. Connect, x0 to x by a path. Covering this pathby polydisks and applying the previous theorem shows that f is constant along it,and therefore f(x) = f(x0).

Corollary 2.7.3. A holomorphic function from X to Cn is constant.

The sheaf of holomorphic sections of the tautological bundle L is denoted byOPn

C(−1) or often simply as O(−1). For any open set U ⊂ PnC viewed as a set of

line, we have

O(−1)(U) = f : U → Cn+1 | f holomorphic and f(`) ∈ `

Lemma 2.7.4. The space of global sections O(−1)(PnC) = 0.

Proof. A global section is given by a holomorphic function f : Pn → Cn+1

satisfying f(`) ∈ `. However f is constant with value, say, v. Thus v ∈⋂` =

0.

A refinement of this given in the exercises will show that a holomorphic linebundle on a compact manifold with at least two sections cannot be the pull backof O(−1).

Similar phenomena hold for algebraic varieties. However, compactness is reallytoo weak a condition when applied to the Zariski topology, since any variety iscompact3. Fortunately, there is a good substitute:

Definition 2.7.5. An algebraic variety X is called complete if and only if forany variety Y , the projection p : X × Y → Y is closed i.e. p takes closed sets toclosed sets.

To why this is an analogue of compactness, we observe:

Proposition 2.7.6. If X is a compact metric space then for any metric spaceY , the projection p : X × Y → Y is closed

Proof. Given a closed set Z ⊂ X × Y and a convergent sequence yi ∈ p(Z),we have to show that the limit y lies in p(Z). By assumption, we have a sequencexi ∈ X such that (xi, yi) ∈ Z. Since X is compact, we can assume that xi convergesto say x ∈ X after passing to a subsequence. Then we see that (x, y) is the limit of(xi, yi) so it must lie in Z because it is closed. Therefore y ∈ p(Z).

We will see in the exercises that positive dimensional affine spaces are notcomplete as we would expect.

3 Since since these spaces are not Hausdorff, this property is usually referred to as quasicom-pactness in most algebraic geometry books, following French terminology.

2.7. COMPACT COMPLEX MANIFOLDS AND VARIETIES 35

Theorem 2.7.7. Projective varieties are complete.

Proof. This can reduced almost immediately to showing that Pnk ×Amk → Amkis closed. The last statement is equivalent to the classical elimination theorem [E,chap 4]: given a collection of polynomials fi(x0, . . . xn, y1, . . . ym) homogeneous inx’s, there exists polynomials gj(y1, . . . ym) such that

∃(x0, . . . xn) 6= 0,∀ifi(x0, . . . ym) = 0⇔ ∀jgj(y1, . . . ym) = 0

See [Har, Mu2, Sh] for alternative proofs.

There is an analogue for proposition 2.7.2.

Proposition 2.7.8. If X is a complete algebraic variety, then all regular func-tions on X are constant.

Proof. Let f be a regular function on X. We can view it as a morphismf : X → A1

k. Since A1k ⊂ P1

k, we also have a morphism F : X → P1k given by

composition. The image f(X) is closed since it coincides with the image of thegraph (x, f(x)) | x ∈ X under projection. Similarly, F (X) is closed. Since Xis irreducible, the same is true of f(X). The only irreducible closed subsets of A1

k

are points and A1k itself. If f(X) = A1

k, then we would be forced to conclude thatF (X) was not closed. Therefore f(X) must be a point.

We can form the sheaf of regular sections of L on Pnk , which we also denotethis by O(−1). This has no nonzero global regular sections by essentially the sameargument as above.

Exercises

1. Show that the function

f(x) =

exp( 1

||x||2−1 ) if ||x|| < 1

0 otherwise

defines a nonzero C∞-function on Rn with support in the unit ball. Con-clude that any C∞-manifold possesses nonconstant C∞-functions.

2. Finish the proof of Theorem 2.7.1.3. If f : X → PnC is a nonconstant holomorphic function between compact

manifolds, prove that f∗OPn(−1) has nonzero holomorphic sections.4. Show that Ank is not complete if n > 0.5. Give a direct proof, without appealing to theorem 2.7.7, that Pnk has no

nonconstant regular functions. Use this to prove that O(−1) has no globalsections.

6. Let G be a projective algebraic group, show that the adjoint representationis trivial.

CHAPTER 3

Basic Sheaf Theory

We introduced sheaves of functions in the previous chapter as a convenientlanguage for defining manifolds and varieties. However, as we will see, there is muchmore to this story. In this chapter, we develop sheaf theory in a more systematicfashion. Presheaves and sheaves are somewhat more general notions than whatwe described earlier. We give the full definitions here, and then explore theirformal properties. We define the notion of an exact sequence in the category ofsheaves. Exact sequences and the associated cohomology sequences, given in thenext chapter, form one of the basic tools used throughout the rest of the book. Wealso give a brief introduction to Grothendieck’s theory of schemes. A scheme isa massive generalization of an algebraic variety, and quite a bit of sheaf theory isrequired just to give the definition.

3.1. The Category of Sheaves

It will be convenient to define presheaves of things other than functions. Forinstance, one might consider sheaves of equivalence classes of functions, distribu-tions and so on. For this more general notion of presheaf, the restrictions mapshave to be included as part the datum:

Definition 3.1.1. A presheaf P of sets (respectively groups or rings) on atopological space X consists of a set (respectively group or ring) P(U) for eachopen set U , and maps (respectively homomorphisms) ρUV : P(U)→ P(V ) for eachinclusion V ⊆ U such that:

1. ρUU = idP(U)

2. ρVW ρUV = ρUW if W ⊆ V ⊆ U .

We will usually write f |V = ρUV (f).

Definition 3.1.2. A sheaf P is a presheaf such that for any open cover Uiof U and fi ∈ P(Ui) satisfying fi|Ui∩Uj

= fj |Ui∩Uj, there exists a unique f ∈ P(U)

with f |Ui = fi.

In English, this says that a collection of local sections can be patched togetherprovided they agree on the intersections.

Definition 3.1.3. Given presheaves of sets (respectively groups) P,P ′ on thesame topological space X, a morphism f : P → P ′ is collection of maps (respectivelyhomomorphisms) fU : P(U)→ P ′(U) which commute with the restrictions. Givenmorphisms f : P → P ′ and g : P ′ → P ′′, the compositions gU fU determinea morphism from P → P ′′. The collection of presheaves of Abelian groups andmorphisms with this notion of composition constitutes a category PAb(X).

37

38 3. BASIC SHEAF THEORY

Definition 3.1.4. The category Ab(X) is the full subcategory of PAb(X) gen-erated by sheaves of Abelian groups on X. In other words, objects of Ab(X) aresheaves, and morphisms are defined in the same way as for presheaves.

A special case of a morphism is the notion of a subsheaf of a sheaf. This is amorphism of sheaves f : P → P ′ where each fU : P(U) ⊆ P ′(U) is an inclusion.

Example 3.1.5. The sheaf of C∞-functions on Rn is a sub sheaf of the sheafof continuous functions.

Example 3.1.6. Let Y be a closed subset of a k-space (X,OX), the ideal sheafassociated to Y ,

IY (U) = f ∈ OX(U) | f |Y = 0,is a subsheaf of OX

Example 3.1.7. Given a sheaf of rings of functions R over X, and f ∈ R(X),the collection of maps R(U)→ R(U) given by multiplication by f |U is a morphism.

Example 3.1.8. Let X be a C∞ manifold, then d : C∞X → E1X is a morphism

of sheaves.

We now introduce the notion of a covariant functor (or simply functor) F :C1 → C2 between categories. This consists of a map F : ObjC1 → ObjC2 and maps

F : HomC1(A,B)→ HomC2(F (A), F (B))

for each pair A,B ∈ ObjC1 such that(F1) F (f g) = F (f) F (g)(F2) F (idA) = idF (A)

In most cases, we will be content to just describe the map on objects, and leave themap on morphisms implicit. There are many examples of functors in mathematics:the map which takes a ring to its group of units, the map which takes a group toits group ring, the map which takes a topological space with a distinguished pointto its fundamental group...

Contravariant functors are defined similarly but with

F : HomC1(A,B)→ HomC2(F (B), F (A))

and the rule (F1) for composition adjusted accordingly. For example, the dualV 7→ V ∗ gives a contravariant functor from the category of vector spaces to itself.

We describe a couple of basic examples involving presheaves.

Example 3.1.9. Let Ab denote the category of Abelian groups. Given a spaceX, the global section functor Γ : PAb(X) → Ab is defined by Γ(P) = Γ(X,P) =P(X) for objects, Γ(f) = fX for morphisms f : P → P ′.

Example 3.1.10. Fix a space X and a point x ∈ X. We define the stalk Px,of P at x, as the direct limit lim−→P(U) over neighbourhoods of x. The elementscan be represented by germs of sections of P, as in Section 2.4. Given a morphismφ : P → P ′, we get an induced map Px → P ′x taking the germ of f to the germ ofφ(f). This gives a functor from PAb(X)→ Ab.

There is a functor generalizing a construction from section 2.1.

Theorem 3.1.11. There is a functor P 7→ P+ from PAb(X) → Ab(X) calledsheafification, with the following properties:

3.1. THE CATEGORY OF SHEAVES 39

(a) If P is a presheaf of functions, then P+ ∼= Ps, where the right side isdefined in example 2.1.10.

(b) There is a canonical morphism P → P+.(c) If P is a sheaf then the morphism P → P+ is an isomorphism (see exercises

for the precise meaning)(d) Any morphism from P to a sheaf factors uniquely through P → P+

(e) The map P → P+ induces an isomorphism on stalks.

Proof. We sketch the construction of P+. We do this in two steps. First,we construct a presheaf of functions P#. Set Y =

∏x∈X Px. We define a sheaf

P# of Y -valued functions and a morphism P → P# as follows. There is a mapσx : P(U)→ Px given by

σx(f) =

the germ fx if x ∈ U0 otherwise

Then f ∈ P(U) determines a function f# : U → Y given by f#(x) = σx(f). Let

P#(U) = f# | f ∈ P(U),

this is yields a presheaf. We have a morphism P(U) → P#(U) given by f 7→ f#.Now apply the construction given earlier in example 2.1.10, to produce a sheafP+ = (P#)s. The composition P(U) → P#(U) ⊂ P+(U), yields the desiredmorphism P → P+. This construction is clearly functorial. If P is already asheaf, then the maps P(U) → P#(U) ⊂ P+(U) can be seen to be isomorphisms.Therefore given a morphism from a presheaf to a sheaf P → F , we have a morphismP+ → F+ ∼= F .

Exercises

1. Let X be a topological space. Fix a symbol ∗. Construct a categoryOpen(X), whose objects are opens subsets of X and let

HomOpen(X)(U, V ) =

∗ if U ⊆ V∅ otherwise

Show that a presheaf of sets (or groups...) on Open(X) is the same thingas a contravariant functor to the category of sets (or groups...).

2. Given a presheaf F ∈ PAb(X) and f ∈ F(X), prove that the supportsupp(f) = x ∈ X | fx 6= 0 is closed.

3. A morphism f : P → Q or presheaves or sheaves is called an isomorphismif there exists a morphism f−1 : Q → P such that f f−1 and f−1 f areboth the identity. Show that f is an isomorphism if and only if each fU isa bijection. If P,Q are sheaves, show that f is an isomorphism if and onlyif each fx : Px → Qx is a bijection.

4. Given an example of a presheaf P which is not isomorphic to P#. Give anexample where P# is not isomorphic to P+.

5. Finish the proof of Theorem 3.1.11.


3.2. Exact Sequences

The categories PAb(X) and Ab(X) are additive which means among otherthings that Hom(A,B) has an Abelian group structure such that composition isbilinear. Actually, more is true. These categories are Abelian [GM, Wl] whichimplies that they possess many of the basic constructions and properties of thecategory of Abelian groups. In particular, there is an intrinsic notion of an exactnessin this category. We give a non intrinsic, but equivalent, formulation of this notionfor Ab(X).

Definition 3.2.1. A sequence of sheaves on X

. . .A → B → C . . .is called exact in the if and only if

. . .Ax → Bx → Cx . . .is exact for every x ∈ X.

The definition of exactness of presheaves will be given in the exercises. We willassume that the symbols A,B, C stand for sheaves unless stated otherwise. We willalso say that morphism A → B is a monomorphism (respectively epimorphism) ifAx → Bx is injective (respectively surjective) for all x ∈ X

Lemma 3.2.2. Let f : A → B and g : B → C, then A → B → C is exact if andonly if for any open U ⊆ X

(1) gU fU = 0.(2) Given b ∈ B(U) with g(b) = 0, there exists an open cover Ui of U and

ai ∈ A(Ui) such that fUi(ai) = b|Ui

.

Proof. We will prove one direction, leaving the other as an exercise. Supposethat A → B → C is exact. To simply notation, we write suppress the subscript U .Given a ∈ A(U), g(f(a)) = 0, since g(f(a))x = g(f(ax)) = 0 for all x ∈ U . Thisshows (1).

Given b ∈ B(U) with g(b) = 0, then for each x ∈ U , bx is the image of agerm in Ax. Choose a representative ai for this germ in some A(Ui) where Ui isa neighbourhood of x. After shrinking Ui if necessary, we have f(ai) = b|Ui . As xvaries, we get an open cover Ui, and sections ai ∈ A(Ui) as required.

Corollary 3.2.3. If A(U) → B(U) → C(U) is exact for every open set U ,then A → B → C is exact.

The converse is false, but we do have:

Lemma 3.2.4. If0→ A→ B → C → 0

is an exact sequence of sheaves, then

0→ A(U)→ B(U)→ C(U)

is exact for every open set U .

Proof. Let f : A → B and g : B → C denote the maps. By lemma 3.2.2,gu fU = 0. Suppose a ∈ A(U) maps to 0 under f , then f(ax) = f(a)x = 0 foreach x ∈ U (we are suppressing the subscript U once again). Therefore ax = 0 foreach x ∈ U , and this implies that a = 0.

3.2. EXACT SEQUENCES 41

Suppose b ∈ B(U) satisfies g(b) = 0. Then by lemma 3.2.2, there exists anopen cover Ui of U and ai ∈ A(Ui) such that f(ai) = b|Ui . Then f(ai|Ui∩Uj −aj |Ui∩Uj ) = 0, which implies ai|Ui∩Uj −aj |Ui∩Uj = 0 by the first paragraph. There-fore ai patch together to yield an element of a ∈ A(U) such that f(a) = b.

We give some natural examples to show that B(X) → C(X) is not usuallysurjective.

Example 3.2.5. Consider the circle S1 = R/Z. Then

0→ RS1 → C∞S1d−→ E1

S1 → 0

is exact. However C∞(S1)→ E1(S1) is not surjective.

To see the first statement, let U ⊂ S1 be an open set diffeomorphic to an openinterval. Then the sequence

0→ R→ C∞(U)f→f ′−→ C∞(U)dx→ 0

is exact by calculus. Thus we get exactness on stalks. The 1-form dx gives a globalsection of E1(S1) since it is translation invariant. However, it is not the differentiala periodic function. Therefore C∞(S1)→ E1(S1) is not surjective.

Example 3.2.6. Let (X,OX) be a C∞ or complex manifold or algebraic varietyand Y ⊂ X a submanifold or subvariety. Let

IY (U) = f ∈ OX(U) | f |Y = 0then this is a sheaf called the ideal sheaf of Y , and

0→ IY → OX → OY → 0

is exact. The map OX(X) → OY (X) need not be surjective. For example, letX = P1

C with OX the sheaf of holomorphic functions. Let Y = p1, p2 ⊂ P1 bea set of distinct points. Then the function f ∈ OY (X) which takes the value 1 onp1 and 0 on p2 cannot be extended to a global holomorphic function on P1 since allsuch functions are constant by Liouville’s theorem.

Given a sheaf S and a subsheaf S ′ ⊆ S, we can define a new presheaf withQ(U) = S(U)/S ′(U) and restriction maps induced from S. In general, this is nota sheaf. We define the quotient sheaf by S/S ′ = Q+.

Exercises

1. Finish the proof of Lemma 3.2.2.2. Give an example of a subsheaf S ′ ⊆ S, where Q(U) = S(U)/S ′(U) fails to

be a sheaf. Check that

0→ S ′ → S → S/S ′ → 0

is an exact sequence of sheaves.3. Given a morphism of sheaves f : S → S ′, define ker f to be the sheafifi-

cation of U 7→ ker fU , and similarly for coker f and im f . Check that thesheafification step is unnecessary for ker f . What about for coker f andim f? Show that at the level of stalks, these operations do what they aresuppose to do.


4. Let Γc(F) ⊆ F(X) consist of sections f whose support supp(f) is compact.Prove that a short exact sequence A → B → C of sheaves, gives an exactsequence 0→ Γc(A)→ Γc(B)→ Γc(C).

5. Define a sequence of presheaves A → B → C to be exact in PAb(X) ifA(U)→ B(U)→ C(U) is exact. Show that if a sequence of sheaves is exactin PAb(X) then it is exact in the sense of 3.2.1, but that converse can fail.

3.3. Direct and Inverse images

Sometimes it is useful to transfer a sheaf from one space to another. Letf : X → Y be a continuous map of topological spaces.

Definition 3.3.1. Given a presheaf F on X, the direct image f∗F is a presheafon Y given by f∗F(U) = F(f−1U) with restrictions

ρf−1Uf−1V : F(f−1U)→ F(f−1V )

Given any subset S ⊂ X of a topological space and a presheaf F , define

(3.3.1) F(S) = lim→F(U)

as U ranges over all open neighbourhoods of S. When S is a point, F(S) is justthe stalk. An element of F(S) can be viewed as germ of section defined in aneighbourhood of S, where two sections define the same germ if there restrictionsagree in a common neighbourhood . From this point of view it is clear that thereis a natural restriction map F(S)→ F(S′) when S′ ⊂ S.

Definition 3.3.2. If G is a presheaf on Y , the inverse image f−1G is a presheafon X given by f−1G(U) = G(f(U)) with restrictions as above.

A closely related operation f∗ will be defined later on. If f : X → Y is theinclusion of a closed set, we also call f∗F extension of F by 0 and f−1G restrictionof G. We sometimes suppress f∗ in the first instance and denote the restriction byG|X .

Lemma 3.3.3. Direct and inverse images of sheaves are sheaves.

Proof. Suppose that f : X → Y is a continuous map and F is a sheaf onX. Let Ui be an open cover of U ⊆ Y , and si ∈ f∗F(Ui) a collection of sectionswhich agree on the intersections. Then f−1Ui is an open cover of f−1U and wecan regard si ∈ F(f−1Ui) as compactible collection of sections for it. Thus we canpatch si to get a uniquely defined s ∈ f∗F(U) = F(f−1U) such that s|Ui

= si. Thisproves that f∗F is a sheaf. The case of inverse images is left as an exercise.

These operations extend to functors f∗ : Ab(X) → Ab(Y ) and f−1 : Ab(Y ) →Ab(X) in an obvious way. While these operations are generally not inverses, there isa relationship. There are canonical morphisms, called adjunctions, α : F → f∗f

−1Fand β : f−1f∗G → G given by the restrictions

F(U)→ F(f(f−1(U))) = f∗f−1F(U), U ⊆ Y

f−1f∗G(V ) = G(f−1(f(V )))→ G(V ), V ⊆ X

Lemma 3.3.4. There is a natural isomorphism

HomAb(X)(f−1F ,G) ∼= HomAb(Y )(F , f∗G)

given by η 7→ (f∗η) α. The inverse is given by ξ 7→ β (f−1ξ).

3.4. THE NOTION OF A SCHEME 43

Proof. Exercise.

In the language of category theory, this says that the functor f−1 is left adjointto f∗.

Definition 3.3.5. Let R be a sheaf of commutative rings (respectively k-algebras) over a space X, the pair (X,R) is called a ringed (respectively k-ringed)space.

For example, any k-space (Section 2.2) is a k-ringed space. The collection ofringed spaces will form a category. To motivate the definition of morphism, observethat from a morphism F of k-spaces (X,R)→ (Y,S), we get a morphism of sheavesof rings S → F∗R given by s 7→ s F .

Definition 3.3.6. A morphism of (k-) ringed spaces (X,R) → (Y,S) is acontinuous map F : X → Y together with a morphism of sheaves of rings (oralgebras) F ′ : S → F∗R.

By the Lemma 3.3.4, to give F ′ is equivalent to giving the adjoint map F−1S →R.

Exercises

1. Finish the proof of Lemma 3.3.3.2. Prove Lemma 3.3.4.3. Give examples where F → f∗f

−1F and f−1f∗G → G are not isomorphisms.4. Generalize Lemma 3.2.4 to show that an exact sequence 0 → A → B →C → 0 of sheaves gives rise to an exact sequence 0 → f∗A → f∗B → f∗C.In other words, f∗ is left exact.

5. Prove that f−1 is exact i.e. that it takes exact sequences to exact sequences.

3.4. The notion of a scheme

Given an ideal I ⊂ k[x0, . . . xn], the set of zeros V (I) depends only on the rad-ical√I = f | ∃n, fn ∈ I. It is clear that we are losing something by considering

just a set of points. Even in classical algebraic geometry, one wanted to considerthe line y = 0 as somehow different from the double line y2 = 0. Grothendieck’stheory of schemes gives a framework in which such things make sense. A schemeis a massive generalization of the notion of an algebraic variety. The canonical ref-erence is [EGA]. Hartshorne’s book [Har] has become the standard introductionto these ideas for most people. Somewhat less austere (and less comprehensive)treatments can be found in [EH, Mu2, Sh]. Since we do not want to give a fullblown treatment of the subject, we will use a much more limited notion scheme inthis book. We will usually refer to these things as just schemes, but occasionally“schemes in our sense” if we need to be clear on what we mean.

Just as the building blocks of general varieties are affine varieties, schemesare built from simpler pieces called affine schemes. Given a commutative R, itsspectrum SpecR is the set of prime ideals of R. It maximal spectrum MspecR ⊂SpecR is the set of maximal ideals.

When R is the coordinate ring of an affine variety Y over an algebraicallyclosed field k, the Hilbert nullstellensatz shows that any maximal ideal of R is ofthe form my = f ∈ R | f(y) = 0 for a unique y ∈ Y . Thus we can identify Y with


MspecR by sending y to my. For this reason, we prefer the maximal spectrum.For the kinds of rings we will working with Mspec will have reasonable properties.However, Spec is the better object in general, and it used as the basis of schemetheory done properly.

For any ideal, I ⊂ R, let

V (I) = p ∈ SpecR | I ⊆ pWe also use V (I) for its intersection with MspecR. The verification of the followingstandard properties will be left as an exercise.

Lemma 3.4.1.(a) V (IJ) = V (I) ∪ V (J).(b) V (

∑Ii) =

⋂i V (Ii),

As a corollary, it follows that the collection of sets of the form V (I) constitutethe closed sets of a topology on SpecR or MspecR called the Zariski topology onceagain. A basis of the Zariski topology on X = SpecR or MspecR is given byD(f) = X − V (f).

Under the identification of an affine variety Y with the maximal spectrum ofits coordinate ring R, V (I) pulls back to the algebraic subset

y ∈ Y | f(y) = 0, ∀f ∈ ISo Y is homeomorphic to MspecR. In fact, we can recover the sheaf of regularfunctions directly from R as well. OY (U) is simply the intersection of the localiza-tions ⋂

m∈UO(Y )m = f ∈ k(Y ) | ∀m ∈ U, f /∈ m.

This description does not generalize well to other rings. So we need an alternativeconstruction.

Proposition 3.4.2. Let R be a commutative ring. There exists a sheaf of ringsOX on X = SpecR such that

(a) OX(D(f)) ∼= R[

1f

]and the restrictions OX(D(f)) → OX(D(gf)) can be

identified with the natural maps R[

1f

]→ R

[1gf

].

(b) The stalk OX,p at p ∈ X is isomorphic to the localization Rp = (R−p)−1R.(c) If R is the coordinate ring of an affine algebraic variety Y = MspecR ⊂ X

then, OX(U) is isomorphic to the ring of regular functions on U ∩ Y .

Proof. Recall that a regular function on an open set U of a variety is anelement of the function field K which can be expressed as g/f with f(x) 6= 0 in aneighbourhood of any x ∈ U . We are going to imitate this definition, but now wemay not have an ambient field since R may not be an integral domain. So we willdefine

OX(U) ⊆∏p∈U

Rp

An element of the product will lie in OX(U) if it is locally given by a fractiong/f with g, f ∈ R. To be more precise notice that we can embed R, and moregenerally R

[1f

], into the product on the right by taking the diagonal embedding

with respect to the maps R[

1f

]→ Rp for p not containing f . An element s ∈

∏Rp

3.4. THE NOTION OF A SCHEME 45

lies in OX(U) if for any p ∈ U , there exists f /∈ p such that s|U∩D(f) lies in the

image of R[

1f

].

By definition, we get maps R[

1f

]→ OX(D(f)). For p ∈ U , the system of

projections OX(U) → Rp induce a map OX,p → Rp. Proving that these areisomorphisms takes some work, and we refer to [Har, pp 71-72] for the details.

Finally, when Y is a variety with coordinate ring R, we can regard Rp as asubring of the function field K = k(Y ). In this way, we can compare definitionsand see that the image of OX(U) ⊂

∏K are just the regular functions on U ∩ Y

diagonally embedded into∏K.

Note that the Zariski topology on MspecR is nothing but the topology in-duced from SpecR. We define OMspecR by restriction: OMspecR(U ∩MspecR) =OSpecR(U). The k-ringed space (MspecR,OMspecR) is called the affine scheme(in our sense) associated to R. We will only consider this construction when Ris a finitely generated k-algebra. In this case, we can express R as a quotientk[x1, . . . xn]/I. As a topological space X = Mspec(R) is the same as V (I) ⊂ An,but the sheaves are generally different. We can give a direct construction of thesheaf, although its debatable whether its any easier to understand. Let

(3.4.1) I(U) = IOAn(U) = f/g | g ∈ k[x1 . . . xn] is nowhere 0 on U, f ∈ I

Then OX is isomorphic to ι−1(OAn/I), where ι is the inclusion X → An. (In thefuture, we will often write OAn/I for this, and the ι−1 is implied.) After unravelingall of this, we see that for an open U ⊂ An, an element of OX(U ∩X) is given bya cover Ui, and a collection of regular functions fi/gi ∈ OAn(Ui), which agree inOAn(Ui ∩ Uj)/IOAn(Ui ∩ Uj).

Definition 3.4.3. A k-scheme (in our sense) is a k-ringed space which is lo-cally isomorphic to an affine scheme (MspecR,OMspecR) with R a finitely generatedk-algebra

A scheme in our sense is equivalent to a Grothendieck scheme locally of finitetype over k. We will usually omit reference to k.

Example 3.4.4. Any prevariety is a scheme.

A scheme X is called seperated if the diagonal ∆X is closed, integral if OX(U)is an integral domain for every U ⊆ X, and of finite type if X is covered by afinite number of (maximal) spectra of finitely generated algebras. For the record,we observe that

Lemma 3.4.5. A variety is the same thing as an integral seperated scheme offinite type.

The simplest example of a scheme which is not a variety is the “fat point”Mspec k[ε]/(ε2). This is a very interesting example, as we will see shortly.

Definition 3.4.6. A morphism of schemes f : (X,OX) → (Y,OY ) is a mor-phism of k-ringed spaces such that the induced maps OY,f(x) → OX,x are local inthe sense that the preimage of mx is mf(x).

For example, morphisms of prevarieties are morphisms of schemes.


Proposition 3.4.7. Let R and S be finitely generated algebras. There is abijection

HomSchemes((MspecS,OMspecS), (MspecR,OMspecR)) ∼= Homk-alg(R,S)

which associates to a morphism F in the left the map on global sections R =OMspecR(MspecR)→ F∗OMspecS(MspecR) = S

Proof. Let X = MspecS and Y = MspecR. Given a homomorphism f :R → S, we just give the construction of the corresponding morphism F : X → Y .The remaining details are very similar to the proof of [Har, chap II, 2.3]. If m ⊂ Sis a maximal ideal, then f−1m is again a maximal ideal by Hilbert’s nullstellensatz.So we can define F (m) = f−1m. It is easy to see that F is continuous. We needto define a morphism OY → F∗OX of sheaves of rings. For any m ∈ MspecS,f induces a local homomorphism fm : RF (m) → Sm by fm(r/t) = f(r)/f(t) fort /∈ m. Thus we get homomorphisms Rn →

∏m∈F−1n Sm and therefore also

fU :∏n∈U

Rn →∏

m∈F−1U

Sm

by taking products of fm. It is easy to see that fU (OY (U)) ⊆ OX(F−1U). Thuswe have our sheaf map OY → F∗OX . The maps on stalks can be identified withthe maps fm which are local. Therefore F is a morphism of schemes. The mapinduced by F on global sections can be identified with f .

Lemma 3.4.8. If X is a scheme, the set of morphisms from Mspec k[ε]/(ε2) toX is in one to one correspondence with the set of vectors in the disjoint union oftangent spaces ∪Tx.

Proof. D = Mspec k[ε]/(ε2) has a unique point m = (ε). We show that the setof morphisms i : D → X with i(m) = x is in one to one correspondence with Tx. Togive such a morphism is equivalent to giving a morphism of D to a neighbourhoodof x. Therefore, we may replace X by an affine scheme MspecR. By the previousproposition, D → MspecR corresponds to a map of algebras i∗ : R → k[ε]/(ε2)such that (i∗)−1m = mx. This factors through R/m2

x. We can see that anyhomomorphism

h : R/m2x = k ⊕mx/m

2x → k[ε]/(ε2) = k ⊕ kε

is given by h(a, b) = a + v(b)ε for v ∈ Tx = (mx/m2x)∗. The argument can be

reversed to show that conversely, any vector in v ∈ Tx determines an algebra mapi∗ : R→ k[ε]/(ε2) for which (i∗)−1m = mx.

A morphism ι : X → Y is called a closed immersion if it is injective and ifOY → OX is an epimorphism. One also says that X is a closed subscheme ofY in this case. We can construct closed subschemes of Pnk fairly concretely. LetI ⊂ k[x0, . . . , xn] be a homogeneous ideal. As a closed set take X = V (I) ⊂ Pnk .Define the sheaf on Pn by

(3.4.2) I(U) = f/g | f ∈ I, g ∈ k[x0, . . . , xn] homogenous of same degree

Then OX is OPn/I. The scheme above depends only the graded ring S/I and isusually denote by Proj S/I. See [Har] for further details.

Exercises

3.5. GLUING SCHEMES 47

1. Prove Lemma 3.4.1.2. For any commutative R ring, check that m ∈ SpecR is closed if and only

if it is a maximal ideal.3. Classify the closed subschemes of MspecR.4. For R = k[x1, . . . xn]/I, show that OMspecR is isomorphic to OAn/I whereI is defined in (3.4.1).

5. Show that (V (I),OPn/I) is a scheme where I is defined in (3.4.2).6. An n-jet on a scheme X is a morphism from Spec k[ε]/(εn+1) to X. Given

an interpretation of these similar to lemma 3.4.8.

3.5. Gluing schemes

Schemes can be constructed by gluing a collection affine schemes together. Thisis similar to giving a manifold, by specifying an atlas for it. Let us describe theprocess explicitly for a pair of affine schemes. Let X1 = MspecR1 and X2 =MspecR2, and suppose we have an isomorphism

φ : R1

[1r1

]∼= R2

[1r2

]for some ri ∈ Ri. Then we have a corresponding diagram of schemes

X2 ⊃ D(r2)Φ∼= D(r1) ⊂ X1

We can define the set X = X1 ∪Φ X2 as the disjoint union modulo the equivalencerelation x ∼ Φ(x) for x ∈ D(r2). We can equip X with the quotient topology, andthen define

OX(U) = (s1, s2) ∈ O(U ∩ U1)×O(U ∩ U2) | φ(s1) = s2Then X becomes a scheme with Xi as an open cover. For example,

R1 = k[x], r1 = x, R2 = k[y], r2 = y, φ(x) = y−1

yields X = P1k. More than two schemes can be handled in a similar way. The data

consists of rings Ri and isomorphisms

φij : Ri

[1rij

]∼= Rj

[1rji

].

These are subject to a compatibility, or cocycle, condition that the isomorphisms

Ri

[1

rijrik

]∼= Rk

[1

rkirkj

]induced by φik and φijφjk coincide, and φii = id, φji = φ−1

ij . The spectra canthen be glued as above. This described in the exercises. For an abstractly givenscheme X to arise from such a construction it suffices to have an affine open coverUi such that Ui ∩Uj is also affine. For this it would be sufficient to assume that Xis seperated and finite type (e.g. a variety). We leave it as an exercise to work thisout explicitly for Pn.

Example 3.5.1. Let f(x0, . . . xn) ∈ k[x0, . . . xn] be a homogeneous polynomialof degree 2d. The double cover of Pn branched along f = 0 is given by

Ri = k[x0i, . . . , xii, . . . , xni, yi]/(yi − f(x01, . . . , 1, . . . xni))


and gluing isomorphisms determined by

(xaj) = (xi/xj)(xai), yj = (xi/xj)dyi

(Things are clearer if we write xai = xa/xi.)

The gluing method can be used to construct normalizations mentioned in thefirst chapter.

Proposition 3.5.2. If X is a variety, then there exists a variety X and amorphism π : X → X such that for any affine open U ⊆ X, OX(π−1U) is theintegral closure of X.

Proof. As a scheme X is integral, seperated and finite type, so it is gotten bygluing a finite number of spectra of integral domains Ri together by isomomorhismφij as above. Let Ri be their integral closures. The φij extend to isomorphisms

φij : Ri

[1rij

]∼= Rj

[1rji

]satisfing the compatability conditions. Thus X can be constructed by gluingSpec Ri. X has finite type by [E, cor. 13.13], and the remaining properties arestraightforward.

Toric varieties are an interesting class of varieties that are explicitly constructedby gluing of affine schemes. The beauty of the subject stems from the interplaybetween the algebraic geometry and the combinatorics. See [F2] for further infor-mation (including an explanation of the name).

To simplify our discussion, we will consider only two dimensional examples.Let 〈, 〉 denote the standard inner product on R2. A strongly convex cone in R2 issubset of the form

σ = t1v1 + t2v2 | ti ∈ R, ti ≥ 0for vectors vi ∈ R2 called generators. It is called rational if the generators can bechosen in Z2. If the generators are nonzero and coincide, σ is called a ray. Thedual cone can be defined by

σ∨ = v | 〈v,w〉 ≥ 0,∀w ∈ σ

In words, it is the intersection of the half planes spanned by the generators. Fix afield k. For each rational strongly convex cone σ, define Sσ to be the subspace ofk[x, x−1, y, y−1] spanned by xmyn for all (m,n) ∈ σ∨ ∩ Z2. This is easily seen tobe a finitely generated subring. The affine toric variety associated to σ is X(σ) =MspecSσ.

A fan ∆ in R2 is a finite collection of non-overlapping rational strongly convexcones.

σ

σ1

2

3.6. SHEAVES OF MODULES 49

Let σi be the collection of cones of ∆. Any two cones intersect in a ray or inthe cone 0. The maps Sσi → Sσi∩σj are localizations at single elements, say sij .This provides us with gluing data

Sσi

[1sij

]∼= Sσj

[1sji

]We define the toric variety X = X(∆) by gluing the above schemes together.

In the example pictured above σ1 and σ2 are generated by (0, 1), (1, 1) and(1, 0), (1, 1) respectively. The varieties

X(σ1) = Mspec k[x, x−1y]

X(σ2) = Mspec k[y, xy−1]are both isomorphic to the affine plane. Set t = x−1y and s = xy−1. Then theseplanes can be glued by identifying (x, t) with (y, s) = (xt, t−1). We will see thisexample again in a different way. It is the blow up of A2 at (0, 0).

Exercises

1. Suppose we are given gluing data as described in the first paragraph. Let ∼be the equivalence relation on

∐MspecRi generated by x ∼ φij(x). Show

X =∐

MspecRi/ ∼ can be made into a scheme with MspecRi as an openaffine cover.

2. Describe Pn by a gluing construction.3. Check that Sσ is a finitely generated algebra when σ is a strongly convex

rational cone.4. Show that the toric variety corresponding to the fan:

(1,0)

(0,1)

(−1, −1)

σ

σ

σ

1

2

3

is P2

3.6. Sheaves of Modules

Definition 3.6.1. Let (X,R) be a ringed space. A subsheaf I ⊂ R is calledan ideal sheaf if I(U) ⊂ R(U) is an ideal for every U .

Example 3.6.2. The sheaf IY ⊂ OX introduced in example 3.2.6 is an idealsheaf.

Example 3.6.3. The affine and projective versions of I in section 3.4 are idealsheaves on Ank and Pnk .


Definition 3.6.4. A sheaf of R-modules or simply an R-module is a sheaf Msuch that each M(U) is an R(U)-module and the restrictions M(U)→M(V ) areR(U)-linear.

An ideal sheaf is sheaf of modules. There are plenty of other natural examples.

Example 3.6.5. Let (X,C∞X ) be a C∞ manifold. Then the space of vector fieldson U ⊂ X is a module over C∞(U) in way compatible with restriction. Thereforethe tangent sheaf TX is a C∞X -module, as is the sheaf of 1-forms E1

X .

Sheaves of modules form a category R-Mod where the morphisms are mor-phisms of sheaves M → N such that each map M(U) → N (U) is R(U)-linear.This is an fact an Abelian category. The notion of exactness in this category coin-cides with the notion introduced in section 3.2.

Before giving more examples, we recall the tensor product and related con-structions [AM, E, Ln].

Theorem 3.6.6. Given modules M1,M2 over a commutative ring R, thereexists an R-module M1 ⊗R M2 and a map (m1,m2) 7→ m1 ⊗m2 of M1 ×M2 →M1 ⊗RM2 which is bilinear and universal:

1. (rm+ r′m′)⊗m2 = r(m⊗m2) + r′(m′ ⊗m2)2. m1 ⊗ (rm+ r′m′) = r(m1 ⊗m) + r′(m1 ⊗m′)3. Any map M1 × M2 → N satisfying the first two properties is given byφ(m1 ⊗m2) for a unique homomorphism φ : M1 ⊗M2 → N .

Definition 3.6.7. Given a module M over a commutative ring R

T ∗(M) = R⊕M ⊕ (M ⊗RM)⊕ (M ⊗RM ⊗RM) . . .

becomes a noncommutative associative R-algebra called the tensor algebra with prod-uct induced by ⊗. The exterior algebra ∧∗M (respectively symmetric algebra S∗M)is the quotient of T ∗(M) by the two sided ideal generated m ⊗ m (respectively(m1 ⊗m2 −m2 ⊗m1)).

The product in ∧∗M is denoted by ∧. We denote by ∧pM (SpM) the submod-ule generated by products of p elements. If V is a finite dimensional vector space,∧pV ∗ (SpV ∗) can be identified with the set of alternating (symmetric) multilinearforms on V in p-variables. After choosing a basis for V , one sees that SpV ∗ aredegree p polynomials in the coordinates.

These operations can be carried over to sheaves:

Definition 3.6.8. Let M and N be two R-modules.1. The direct sum M⊕N is the sheaf U 7→ M(U)⊕N (U).2. The tensor productM⊗N is the sheafification of the presheaf U 7→ M(U)⊗R(U)

N (U).3. The dualM∗ ofM is sheafification of the presheaf U 7→ HomR|U (M|U ,R|U ).4. The pth exterior power ∧pM is sheafification of U 7→ ∧pM(U).5. The pth symmetric power SpM is sheafification of U 7→ SpM(U).

We will see in the exercises that the sheafification step in the definition of M∗is unnecessary. We have already encountered this construction implicitly, the sheafof 1-forms E1

X on a manifold is the dual of the tangent sheaf TX . The objects ofclassical tensor analysis are sections of the sheaf T ∗(TX ⊕ E1

X). A special case offundamental importance is:

3.6. SHEAVES OF MODULES 51

Definition 3.6.9. The sheaf of p-forms on a C∞-manifold is EpX = ∧pE1X .

We want to describe some examples in the algebraic setting. First we describesome standard methods for transferring modules between rings. If f : R → S isa homomorphism of commutative rings, then any R-module M gives rise to anS-module S ⊗R M , with S acting by s(s1 ⊗m) = ss1 ⊗m. This construction iscalled extension of scalars. In the opposite direction, an S-module can always beregarded as an R-module via f . This is restriction of scalars.

Example 3.6.10. Let R be a finitely generated k-algebra, and M an R-module.Let X = MspecR. We define an OX-module M by extending scalars M(U) =OX(U)⊗RM . Such a module is called quasi-coherent. When M = I is an ideal ina polynomial ring, this coincides with the ideal sheaf constructed in (3.4.1).

It will be useful to observe:

Lemma 3.6.11. The functor M 7→ M is exact, i.e. if

0→M1 →M2 →M3 → 0

is exact, then so is0→ M1 → M2 → M3 → 0

Proof. This follows from the fact that OX(U) is a flat R-module. We willdefer the discussion of this notion until Section 17.2.

A final source of examples will come from vector bundles. These examples havea special property.

Definition 3.6.12. A module M over (X,R) is locally free (of rank n) if forevery point has a neighbourhood i : U → X, such that i−1M is isomorphic to afinite (n-fold) direct sum i−1Rn = i−1R⊕ . . .⊕ i−1R. (We will usually write M|Uinstead of i−1M in the future.)

Let π : V → X be a rank n C∞ real (or holomorphic or algebraic) vector bundleon a manifold or variety (X,R) with a local trivialization (Ui, φi), as discussedin §2.6 Recall that we have a sheaf

V(U) = s : U → π−1U is a morphism |π s = idU

Via the trivialization, V(U) can identified with a submodule of∏iR(Ui ∩ U)n. In

this way, V becomes a sheaf of R-modules. The map φi induces an isomorphismV|Ui

∼= R|nUi. Thus V is locally free of rank n. Conversely, we will see in Section

7.3 that every such sheaf arises this way.Locally free modules over certain affine schemes have a simple characterization:

Theorem 3.6.13. Let M be a finitely generated module over a Noetherian com-mutative integral domain R. The following are equivalent

(a) M is locally free.(b) Mp is free for all p ∈ MspecR.(c) M is projective i.e. a direct summand of a finitely generated free module.(d) The function p 7→ dim k(p) ⊗R M on MspecR is constant, where k(p) is

the quotient field of R/p.


Proof. See [E, 19.2] for the equivalence of (a), (b), (c). We prove the equiva-lence of (b) and (d). Let K be the quotient field of R. Note that k(p) is the residuefield of Rp. If Mp is free, then k(p)⊗RM = k(p)⊗RMp and K ⊗RM = K ⊗RMp

have the same dimension. So the dimension function is constant. Suppose con-versely that the dimension is constant. Then dim k(p)⊗RMp = dimK ⊗RM = n.Choose a set of elements m1, . . .mn which maps onto a basis of k(p)⊗RMp. Thenthese generate Mp by Nakayama’s lemma [E, cor. 4.8]. So we get a surjectionφ : Rnp →Mp. We will check that kerφ = 0, which will imply that Mp is free. SinceRp is a domain, it suffices to check this after tensoring by K. After doing so, we geta surjection of n dimensional vector spaces, which is necessarily an isomorphism.

Suppose that f : (X,R) → (Y,S) is a morphism of ringed spaces. Givenan R-module M, f∗M is naturally an f∗R-module, and hence an S-module byrestriction of scalars via S → f∗R. Similarly given an S-module N , f−1N isnaturally an f−1S-module. We define the R-module

f∗N = R⊗f−1S f−1N

where the R is regarded as a f−1S-module under the adjoint map f−1S → R.When f is injective, we often write N|X instead of f∗N .

The inverse image of a locally free sheaf is easily seen to be locally free. Thishas an interpretation in the context of vector bundles. If π : V → Y is a vectorbundle, then the pullback f∗V → X is the vector bundle given set theoretically asthe projection

f∗V = (v, x) |π(v) = f(x) → X

Thenf∗(V) ∼= (sheaf of sections of f∗V )

A special case is when X = y → Y is the inclusion of a point. Then the fiberπ−1y can be identified with f∗(V) = (Ry/my)⊗ Vy.

Set P = PnC and V = Cn+1. Recall that the tautological line bundle on projec-tive space P is given by

L = (x, `) ∈ V × P | x ∈ `

with its natural projection to P. The sheaf of holomorphic sections of L is denotedby OP(−1) = OPn

C(−1). Let OP(1) = OP(−1)∗. A more geometric description will

be given later in exercises of Section 7.3. Let U = Pn+1 − p, and let π : U → Pbe a linear projection. Then we will see that π : U → Pn is a line bundle, and thatOP(1) is its sheaf of sections. For each integer m, let

OP(m) =

O(1)⊗ . . .⊗O(1) (m times) if m > 0OP if m = 0O(−1)⊗ . . .⊗O(−1) (−m times) if m < 0

These are rank one locally free sheaves, and we shall see that all such sheaves on Pare of this form.

We have an inclusion of L into the trivial vector bundle V × P. This gives riseto a morphism of locally free sheaves

0→ OP(−1)→ OP ⊕ . . .⊕OP ∼= V ⊗C OP

3.7. DIFFERENTIALS 53

Dualizing and taking symmetric powers yields

SmV ∗ ⊗OP → SmOP(1)

whenm ≥ 0. We have a canonical mapOP(m)→ SmOP(1) which is an isomorphismon stalks, and hence an isomorphism. Combining these and applying proposition2.7.2 gives maps

SmV ∗ → Γ(OP(m))We will see later that these maps are isomorphisms (in the algebraic case as well).Thus the global sections of OP(m) are homogeneous degree m polynomials in thehomogeneous coordinates of m.

Exercises

1. Show that M really is a sheaf, and that its stalk at p is precisely thelocalization Mp.

2. Show that direct sums, tensor products, exterior, and symmetric powers oflocally free sheaves are locally free.

3. Given two R-modules M,N , define the presheaf Hom(M,N ) by U 7→HomR|U (M|U ,N|U ). Show that this is a sheaf. Note thatM∗ = Hom(M,R).

4. Given an R-module M over X, the stalk Mx is an Rx-module for anyx ∈ X. If M is a locally free R-module, show that then each stalk Mx isa free Rx-module of finite rank. Show that the converse can fail.

5. Suppose that X is an affine algebraic variety. Let M be a finitely generatedO(X)-module, show that there exists a nonempty open set U ⊆ X suchthat M |U is locally free.

6. Verify the assertion f∗(V) ∼= (sheaf of sections of f∗V ) made above.

3.7. Differentials

With basic sheaf theory in hand, we can now construct sheaves of differentialforms on manifolds and varieties in a unified way. In order to motivate things, let usstart with a calculation. Suppose that X = Rn with coordinates x1, . . . xn. Givena C∞ function f on X, we can develop a Taylor expansion about (y1, . . . yn):

f(x1, . . . xn) = f(y1, . . . yn) +∑ ∂f

∂xi(y1, . . . yn)(xi − yi) +O((xi − yi)2)

Thus the differential is given by

df = f(x1, . . . xn)− f(y1, . . . yn) mod (xi − yi)2.

We can view x1, . . . xn, y1, . . . yn as coordinates on X×X = R2n, so that xi−yi = 0defines the diagonal ∆. Then df lies in the ideal of ∆ modulo its square.

Let X be a C∞ or complex manifold or an algebraic variety over a field k. Wetake k = R or C in the first two cases. We have diagonal map δ : X → X×X givenby x 7→ (x, x), and projections pi : X ×X → X. Let I∆ be the ideal sheaf of theimage of δ, and let I2

∆ be the submodule locally generated by products of pairs ofsections of I∆. Then we define the sheaf of 1-forms by

Ω1X = I∆/I

2∆.

This has two different OX -module structures, we pick the first one where OX actson I∆ through p−1

1 OX → OX×X . We define the sheaf of p-forms by ΩpX = ∧pΩ1X .


We define a morphism d : OX → Ω1X of sheaves (but not OX -modules) by df =

p∗1(f)− p∗2(f). This has the right formal properties because of the following:

Lemma 3.7.1. d is a k-linear derivation

Proof. By direct calculation

d(fg)− fdg − gdf = [p∗1(f)− p∗2(f)][p∗2(g)− p∗2(g)] ∈ I2∆.

Differentials behave contravariantly with respect to morphisms. Given a mor-phism of manifolds or varieties f : Y → X, we get a morphism Y × Y → X ×Xpreserving the diagonal. This induces a morphisms of sheaves g∗Ω1

X → Ω1Y which

fits into a commutative square

g∗OX → OYd ↓ ↓ dg∗Ω1

X → Ω1Y

The calculations at the beginning of this section basically show that:

Lemma 3.7.2. If X is C∞ manifold, then Ω1X∼= E1

X and d coincides the deriv-ative constructed in Section 2.6.

Since complex manifolds are also C∞-manifolds, there is potential for confu-sion. So in the sequel, we will only use the symbol ΩpX in the holomorphic oralgebraic case. Sections of E1

X over a coordinate neighbourhood are given by sums∑fidxi with C∞-coefficients. A similar statement holds for sections of Ω1

X oncomplex manifolds, except that xi should be chosen as analytic coordinates, andthe coefficients should be holomorphic.

Now suppose thatX ⊂ ANk be a closed subvariety defined by the ideal (f1, . . . fr).Let R = k[x1, . . . xn]/(f1, . . . fr) be the coordinate ring. Then Ω1

X is quasi-coherent.In fact, it is given by ΩR/k where the module of Kahler differentials [E, chap. 16]

ΩR/k =ker[R⊗k R→ R]ker[R⊗k R→ R]2

Using standard exact sequences [loc. cit, 16.2] or [Har, II.8], we arrive at morecongenial description of this module.

ΩR/k ∼=⊕

`Rdx`∑iRdfi

∼= coker(∂fi/∂xj)

Theorem 3.7.3. X is nonsingular if and only if Ω1X is locally free of rank equal

to dimX.

Proof. X is nonsingular if and only if the function x 7→ dim k(mx) ⊗ Ω1R/k

takes constant value dimX. So the result follows from theorem 3.6.13.

Exercises

1. When X is a manifold or variety, the tangent sheaf TX = (Ω1X)∗. Show that

a section D ∈ TX(U) determines an OX(U)-linear derivation from OX(U)to OX(U).

3.7. DIFFERENTIALS 55

2. Given subvariety or submanifold i : X ⊂ Y . Show that there is an epimor-phism i∗Ω1

Y = OY ⊗Ω1Y → Ω1

X . Describe the kernel when X is defined bya single equation F = 0.

3. Prove that M ∼= M∗∗ if M is locally free. Show that Ω1X need not be

isomorphic to T ∗X if X is singular. Thus Ω1X contains more information in

general.

CHAPTER 4

Sheaf Cohomology

As we saw earlier a “surjection” or epimorphism F → G of sheaves, need notinduce a surjection of global sections. We would like to understand what furtherconditions are required to ensure this. This may seem like a fairly technical problem,but it lies at the heart of many fundamental questions in geometry and functiontheory. Typically, we may want to know when some interesting class of functionsextend from a subspace to the whole space, and this is a special case of the aboveproblem.

In this chapter, we introduce sheaf cohomology which gives a framework inwhich to answer such questions and more. Our basic approach is due to Godement[G], but we have set things up as an inductive definition. This will allow us to getto the main results rather quickly.

4.1. Flasque Sheaves

Definition 4.1.1. A sheaf F on X is called flasque (or flabby) if the restrictionmaps F(X)→ F(U) are surjective for any nonempty open set U .

The importance of flasque sheaves stems from the following:

Lemma 4.1.2. If 0→ A→ B → C → 0 is an exact sequence of sheaves with Aflasque, then B(X)→ C(X) is surjective.

Proof. We will prove this by the no longer fashionable method of transfiniteinduction1. Let γ ∈ C(X). By assumption, there is an open cover Uii∈I , suchthat γ|Ui

lifts to a section βi ∈ B(Ui). By the well ordering theorem, we can assumethat the index set I is the set of ordinal numbers less than a given ordinal κ. Wewill define

σi ∈ B(⋃j<i

Uj)

inductively, so that it maps to the restriction of γ. Set σ1 = β0. Now supposethat σi exists. Let U = Ui ∩ (∪j<iUj). Then βi|U − σi|U is the image of a sectionα′i ∈ A(U). By hypothesis α′i extends to a global section αi ∈ A(X). Then set

σi+1 =

σi on

⋃j<i Uj

βi − αi|Ui on Ui

If i is a limit (non-successor) ordinal, then the previous σj ’s patch to define σi.Then σκ is a global section of B mapping to γ.

1For most cases of interest to us, X will have a countable basis, so ordinary induction willsuffice

57

58 4. SHEAF COHOMOLOGY

Corollary 4.1.3. The sequence 0 → A(X) → B(X) → C(X) → 0 is exact ifA is flasque.

Proof. Combine this with lemma 3.2.4.

Example 4.1.4. Let X be a space with the property that any open set is con-nected (e.g. if X is irreducible). Then any constant sheaf is flasque.

Let F be a presheaf, define the presheaf G(F) of discontinuous sections of F ,by

U 7→∏x∈UFx

with the obvious restrictions. There is a canonical morphism F → G(F), sendingf ∈ F(U) to the product of germs (fx)x∈U .

Lemma 4.1.5. G(F) is a flasque sheaf, and the morphism F → G(F) is amonomorphism if F is a sheaf.

G clearly determines a functor from Ab(X) to itself.

Lemma 4.1.6. G is an exact functor i.e. it preserves exactness.

Proof. Given an exact sequence 0 → A → B → C → 0, the sequence 0 →Ax → Bx → Cx → 0 is exact by definition. Therefore

0→∏x∈UAx →

∏x∈UBx →

∏x∈UCx → 0

is exact. Therefore0→ G(A)→ G(B)→ G(C)→ 0

is an exact sequence of presheaves, and hence of sheaves (by the exercises of Section3.2).

Let Γ : Ab(X) → Ab denote the functor of global sections, Γ(X,F) = Γ(F) =F(X).

Lemma 4.1.7. Then Γ G : Ab(X)→ Ab is exact.

Proof. Follows from corollary 4.1.3 and Lemma 4.1.6.

Exercises

1. Find a proof of Lemma 4.1.2 which uses Zorn’s lemma.2. Prove Lemma 4.1.5.3. Let Xdisc denote the set X with its discrete topology, and let π : Xdisc → X

be the identity map. Show that G(F) = π∗π∗F , and that the above map

F → G(F) is the adjunction map (§3.3).4. Prove that the sheaf of bounded continuous real valued functions on R is

flasque5. Prove the same thing for the sheaf of bounded C∞ functions on R.6. Prove that if 0→ A → B → C is exact and A is flasque, then 0→ f∗A →f∗B → f∗C → 0 is exact for any continuous map f .

4.2. COHOMOLOGY 59

4.2. Cohomology

Let F be a sheaf on a topological space X. Define the following operations byinduction

Ci(F) =

F if i = 0coker[F → G(F)] if i = 1C1Cn(F) if i = n+ 1

where coker[F → G(F)] is the sheaf associated to the presheaf

U 7→ coker[F(U)→ G(F)(U)].

The map F → G(F) is the canonical map constructed in the previous section. Bydefinition, this can be prolonged to an exact sequence

0→ F → G(F)→ C1(F)→ 0

Thus we also have exact sequences

0→ Cn(F)→ G(Cn(F))→ Cn+1(F)→ 0

Sheaf cohomology can now be defined inductively:

Definition 4.2.1.

Hi(X,F) =

Γ(X,F) if i = 0coker[Γ(X,G(F))→ Γ(X,C1(F))] if i = 1H1(X,Cn(F)) if i = n+ 1

The following can be deduced from the definition by induction.

Lemma 4.2.2. F 7→ Hi(X,F) is a functor from Ab(X)→ Ab.

We now come to the key result.

Theorem 4.2.3. Given an exact sequence of sheaves

0→ A→ B → C → 0,

there is a long exact sequence

0→ H0(X,A)→ H0(X,B)→ H0(X, C)→ H1(X,A)→ H1(X,B)→ . . .

The proof will be based on a couple of lemmas.

Lemma 4.2.4. (Snake lemma) Given commutative diagram of Abelian groups

0 // A1//

α

B1//

β

C1//

γ

0

0 // A2// B2

// C2

with exact rows, there is a canonical exact sequence

0→ ker(α)→ ker(β)→ ker(γ)→ coker(α)→ coker(β)→ coker(γ)

The last map is surjective if B2 → C2 is.

The existence of the sequence is standard, and can be proved by a diagramchase (or looked up in many places [AM, E, Ln]). The only thing we want toexplain is how the sequence can be made canonical, since this is not always clearfrom standard proofs, and canonicity is crucial for us.


Proof. Most of the maps are the obvious ones, and these are clearly natural.We explain how to construct a canonical connecting (or “snaking”) homomorphismδ : ker(γ)→ coker(α). Let f : B1 → C1 be the given map. Define P = f−1 ker(γ) ⊆B1 and let K be the kernel of the surjective homomorphism P → ker(γ). We notethat f induces an isomorphism F : P/K ∼= ker(γ). The image of P under β mapsto 0 in C2. Therefore, we have a canonical map δ′ : P → A2 through which β|Pfactors. A simple chase shows that δ′(K) ⊆ im(α). Therefore we get an inducedmap

δ : ker(γ) F−1

−→ P/Kδ′−→ coker(α)

We can now extend this to sheaves.2

Lemma 4.2.5. (Snake lemma II) Given commutative diagram of sheaves Abeliangroups

0 // A1//

α

B1//

β

C1 //

γ

0

0 // A2// B2

// C2with exact rows, there is a canonical exact sequence

0→ ker(α)→ ker(β)→ ker(γ)→ coker(α)→ coker(β)→ coker(γ)

The last map is an epimorphism if B2 → C2 is.

Proof. Since we have canonical maps in previous lemma, these extend tosheaves. The exactness can checked on stalks, and so this reduces to the previousstatement.

Lemma 4.2.6. There is a commutative diagram with exact rows and columns

0

0

0

0 // A

// B

// C

// 0

0 // G(A)

// G(B)

// G(C)

// 0

0 // C1(A)

// C1(B)

// C1(C)

// 0

0 0 0

2Although we won’t appeal to it explicitly, it is worth pointing out a remarkable metatheorem

that any result about finite diagrams true in for all categories of modules is true of any Abeliancategory, and in particular for sheaves of Abelian groups. This follows from the work of Freyd,

Mitchell...

4.2. COHOMOLOGY 61

Proof. By Lemma 4.1.6, there is a commutative diagram with exact rows

0

0

0

0 // A

// B

// C

// 0

0 // G(A) // G(B) // G(C) // 0

The snake lemma gives the rest.

Proof of theorem. From the Lemmas 3.2.4, 4.1.7, 4.2.6 we get a commu-tative diagram with exact rows:

0 → Γ(G(A)) → Γ(G(B)) → Γ(G(C)) → 0↓ ↓ ↓

0 → Γ(C1(A)) → Γ(C1(B)) → Γ(C1(C))

From the snake lemma, we obtain a 6 term exact sequence

0→ H0(X,A)→ H0(X,B)→ H0(X, C)

→ H1(X,A)→ H1(X,B)→ H1(X, C).which we need to extend. Applying this argument to

0→ C1(A)→ C1(B)→ C1(C)→ 0

yields a 6 term sequence

. . . H0(X,C1(B))→ H0(X,C1(C)) δ→ H2(X,A)→ . . .

which is not quite what we want. We will show that the connecting map δ factorsthrough a map δ of

H1(X, C) =H0(X,C1(C))H0(X,G(C))

Given an element γ ∈ H1(X, C), suppose we chose two different lifts γ, γ′ ∈H0(X,C1(C)). Then γ−γ′ lies in H0(X,G(C)), which can be lifted to H0(X,G(B))by Lemma 4.1.7. An easy diagram chase using

H0(G(B)) //

H0(G(C)) //

0

H0(C1(B)) //

H0(C1(C)) δ //

H2(A)

H1(B) // H1(C)

δ

99rr

rr

r

shows that δγ = δγ′ as desired. So that δγ = δγ is well defined. We also seeimmediately, that δγ = 0 if and only if γ lies in the image of H1(X,B). Thus wehave a 6 term cohomology sequence

. . . H1(X,B)→ H1(X, C) δ→ H2(X,A)→ . . .


which extends the earlier one. Applying this to

0→ Ci(A)→ Ci(B)→ Ci(C)→ 0

shows that we can continue this sequence indefinitely.

Corollary 4.2.7. B(X)→ C(X) is surjective if H1(X,A) = 0.

We make an addendum to the theorem that says that cohomology sequence isnatural, or in words that the Hi(X,−) form a “δ-functor” [Gr, Har]. By a mapor morphism of diagrams (and particular exact sequences), we will simply mean amap between objects in the same position which commute with the maps of thediagram.

Theorem 4.2.8. A morphism of short exact sequences of sheaves gives rise toa map of long exact sequences of cohomology.

Exercises

1. If F is flasque prove that Hi(X,F) = 0 for i > 0. (Prove this for i = 1,and that F flasque implies that C1(F) is flasque.)

2. Finish all the details in the proof of the snake lemmas.3. Give a proof of Theorem 4.2.8. (Use as much paper and coffee as necessary.)4. Given a sheaf A on X, let Ext1(Z,A) denote the set of exact sequences

0→ A→ B → Z→ 0

modulo the equivalence relation that two sequences are exact if and only ifthey fit into a commutative diagram

0 // A //

=

B //

Z //

=

0

0 // A // C // Z // 0

Check that this is in fact an equivalence relation. Show that the classδ(1) ∈ H1(X,A) associated to an exact sequence determines a well definedmap Ext1(Z,A)→ H1(X,A). Show that this is a bijection.

4.3. Soft sheaves

We introduce the class of soft sheaves, whose definition is similar to the classflasque sheaves. However, it is a larger class, and thus more convenient. We assumethroughout this section that X is a metric space, although the results hold underthe weaker assumption that X is paracompact and Hausdorff.

Definition 4.3.1. A sheaf F is called soft if the map F(X)→ F(S) is surjec-tive for all closed sets, where F(S) was defined in (3.3.1).

Lemma 4.3.2. If 0→ A→ B → C → 0 is an exact sequence of sheaves with Asoft, then B(X)→ C(X) is surjective.

Proof. The proof is very similar to the proof of lemma 4.1.2. We just indicatethe modifications. Let γ ∈ C(X). We can assume that the open cover Ui | i < κ,

4.3. SOFT SHEAVES 63

such that γ|Uilifts to B, consists of open balls. Let Vi be a new open cover where

we shrink the radii of each ball, so that Vi ⊂ Ui. We can define

σi ∈ B(∪j<iVj)

inductively as before, so that σκ maps to γ.

Corollary 4.3.3. If A and B are soft then so is C.

Proof. For any closed set S, B(X) → B(S) is surjective. The lemma showsthat B(S) → C(S) is surjective. Therefore B(X) → C(S) is surjective, and thisimplies the same for C(X)→ C(S).

One trivially has:

Lemma 4.3.4. A flasque sheaf is soft.

Lemma 4.3.5. If F is soft then Hi(X,F) = 0 for i > 0.

Proof. Lemma 4.3.2 applied to 0 → F → G(F) → C1(F) → 0 shows thatH1(F) = 0. Corollary 4.3.3 and the previous lemma imply that C1(F) is soft. Byinduction it follows that all the Ci(F) are soft. Hence Hi(F) = 0 for all i.

Theorem 4.3.6. The sheaf ContX,R of continuous real valued functions on ametric space X is soft.

Proof. The basic strategy of this proof and many of the subsequent softnessproofs is the construction of a continuous “cutoff” function ρ which is 0 outsidea given neighbourhood U of a closed set S, and 1 close to S. Then given anycontinuous function f : U → R, ρf can be extended by 0 to all of X. Since f andρf have the same germ along S, this would prove the surjectivity of ContX,R(X)→ContX,R(S) as required

To construct ρ we proceed as follows. Let S1 ⊂ U be a closed set containing Sin its interior. This can be constructed by expressing U as a union of open balls,and taking the union of closed balls of half the radii. Let S2 = X − U . ThenUrysohn’s lemma [Ke] guarantees the existence of a continuous ρ, taking a valueof 1 on S1 and 0 on S2.

We get many more examples of soft sheaves with the following.

Lemma 4.3.7. Let R be a soft sheaf of rings, then any R-module is soft.

Proof. Let m be section of an R-module defined in the neighbourhood of aclosed set S, and let S2 be the complement of this neighbourhood . Since R is soft,the section which is 1 on S and 0 on S2 extends to a global section ρ. Then ρmextends to a global section of the module.

Let S1 ⊂ C denote the unit circle, and let e : R → S1 denote the normalizedexponential e(x) = exp(2πix). Let us say that X is locally simply connected if everyneighbourhood of every point contains a simply connected neighbourhood.

Lemma 4.3.8. If X is locally simply connected, then the sequence

0→ ZX → ContX,Re−→ ContX,S1 → 1

is exact.


Proof. Exactness of the sequence of sections can be checked in any simplyconnected neighbourhood of a point. This implies exactness of the sequence stalks.

Lemma 4.3.9. If X is simply connected and locally simply connected, thenH1(X,ZX) = 0.

Note that in the future, we will usually write Hi(X,Z) instead of Hi(X,ZX),and likewise for other constant sheaves.

Proof. Since X is simply connected, any continuous map from X to S1 can belifted to a continuous map to its universal cover R. In other words, CR(X) surjectsonto CS1(X). Since CR is soft, Lemma 4.3.8 implies that H1(X,Z) = 0.

Corollary 4.3.10. H1(Rn,Z) = 0.

Exercises

1. Show that ContR,R is not flasque.2. Show that a module over a flasque sheaf of rings need not be flasque.3. Suppose that X is a locally compact metric space. If 0→ A→ B → C → 0

is an exact of sheaves with A soft, show that Γc(B) → Γc(C) is surjective,where Γc are the sections with compact support.

4. Assuming X is locally compact, we can define cohomology with com-pact support by setting H0

c (X,F) = Γc(F), H1c (X,F) = coker[Γc(F) →

Γc(C1(F))] and so on. Construct a long exact sequence (or at least thefirst 6 terms) analagous to theorem 4.2.3.

5. Again assuming X is locally compact, prove that the higher cohomologywith compact support of a soft sheaf is trivial.

4.4. C∞-modules are soft

We want prove that the sheaf of C∞ functions on a manifold is soft. We startwith a few lemmas.

Lemma 4.4.1. Given ε > 0, there exists an R-linear “smoothing” operatorΣε : C(Rn) → C∞(Rn) such that if f(x) is constant for all points x in an ε ballabout x0, then Σε(f)(x0) = f(x0).

Proof. Let ψ be a C∞ function on Rn with support in ||x|| < 1 such that∫ψdx = 1. Rescale this by setting φ(x) = εnψ(x/ε). Then

Σε(f)(x) =∫

Rn

f(y)φ(x− y)dy

will have the desired properties.

Lemma 4.4.2. Let S ⊂ Rn be a compact subset, and let U be an open neigh-bourhood of S, then there exists a C∞ function ρ : Rn → R which is 1 on S and 0outside U .

Proof. We constructed a continuous function ρ1 : Rn → R with these prop-erties in the previous section. Then set ρ = Σε(ρ1) with ε sufficiently small.

4.5. MAYER-VIETORIS SEQUENCE 65

We want to extend this to a manifold X. For this, we need the followingconstruction. Let Ui be a locally finite open cover of X, which means that everypoint of X is contained in a finite number of Ui’s. A partition of unity subordinateto Ui is a collection C∞ functions φi : X → [0, 1] such that

1. The support of φi lies in Ui.2.∑φi = 1 (the sum is meaningful by local finiteness).

Partitions of unity always exist for any locally finite cover, see [Sk2, Wa] and theexercises.

Lemma 4.4.3. Let S ⊂ X be a closed subset, and let U be an open neighbourhoodof S, then there exists a C∞ function ρ : X → R which is 1 on S and 0 outside U .

Proof. Let Ui be a locally finite open cover of X such that each Ui isdiffeomorphic to a ball Rn (see the exercises). Then we have functions ρi ∈ C∞(Ui)which are 1 on S∩Ui and 0 outside U∩Ui by the previous lemma. Choose a partitionof unity φi. Then ρ =

∑φiρi will give the desired function.

Theorem 4.4.4. Given a C∞ manifold X, C∞X is soft.

Proof. Given a function f defined in a neighbourhood of a closed set S ⊂ X.Let ρ be a given as in the Lemma 4.4.3, then ρf gives a global C∞ functionextending f .

Corollary 4.4.5. Any C∞X -module is soft.

These arguments can be extended by introducing the notion of a fine sheafdescribed below.

Exercises

1. Let X be an n dimensional C∞ manifold. Prove that it has a locally finiteopen cover Ui such that each Ui is diffeomorphic to Rn.

2. Let X and Ui be as in the previous exercise. Construct a partition ofunity in this case, by first constructing a family of continuous functionssatisfying these conditions, and then applying Σε.

3. A sheaf F is called fine if for each locally finite cover Ui there exists apartition of unity which is a collection of endomorphisms φi : F → F suchthat(a) The endomorphism of Fx induced by φi vanishes for x /∈ Ui.(b)

∑φi = 1.

Prove that a fine sheaf on a paracompact Hausdorff space is soft.4. Assuming the exercises of the previous section, compute H∗c (R,R) by us-

ing the complex 0 → R → C∞R → C∞R → 0, where the last map is thederivative.

4.5. Mayer-Vietoris sequence

Given a continuous map f : Y → X and a sheaf F ∈ Ab(X), we defined theinverse f−1F ∈ Ab(Y ) so that

Γ(Y, f−1F) = F(f(Y )) = lim−→U⊇f(Y )

F(U)


Thus we have a natural map

p0 : Γ(X,F)→ Γ(Y, f−1F)

induced from restriction.

Theorem 4.5.1. There exists natural maps pi,F : Hi(X,F) → Hi(Y, f−1F),agreeing with above map when i = 0, such that a short exact sequence

0→ F1 → F2 → F3 → 0

gives rise to a commutative diagram

. . . // Hi(X,F1) //

Hi(X,F2) //

Hi(X,F3) //

. . .

// Hi(Y, f−1F1) // Hi(Y, f−1F2) // Hi(Y, f−1F3) //

Proof. We define the map by induction. For i = 0 the map p0 has alreadybeen defined. We construct a morphism f−1G(F) → G(f−1F) as follows. LetU ⊂ Y , then let

β : f−1G(F)(U) =∏

x∈f(U)

Fx →∏y∈UFf(y) = G(f−1F)(U)

be defined as the identity Fx → Ff(y) if x = f(y), and 0 otherwise. Then there isa unique morphism

α1 : f−1Ci(F)→ Ci(f−1F)fitting into a commutative diagram.

0 // f−1F //

id

f−1G(F) //

β

f−1C1(F) //

α1

0

0 // f−1F // G(f−1F) // C1(f−1F) // 0

We can iterate this to obtain a morphism

αi : f−1Ci(F)→ Ci(f−1F)

Then from what has been defined so far, we get the diagram below, marked withsolid arrows.

Γ(X,G(F)) //

Γ(X,C1(F)) //

H1(X,F)

p1

// 0

Γ(Y,G(f−1F)) // Γ(Y,C1(f−1F)) // H1(Y, f−1F) // 0

This gives rise to the dotted arrow p1. For i > 1, we define pi as the composite

H1(X,Ci−1(F))p1−→ H1(Y, f−1Ci−1(F))

αi−1−→ H1(Y,Ci−1(f−1F))

The remaining properties are left as an exercise.

When f : Y → X is an inclusion of an open set, we will usually write Hi(Y,F)for Hi(Y, f−1F) and refer the induced map on cohomology as restriction. We willintroduce a basic tool for computing cohomology groups which is a prelude to Cechcohomology.

4.5. MAYER-VIETORIS SEQUENCE 67

Theorem 4.5.2. Let X be a union of two open sets U ∪V and let F be a sheafon X. Then there is a long exact sequence

. . . Hi(X,F)→ Hi(U,F)⊕Hi(V,F)→ Hi(U ∩ V,F)→ Hi+1(X,F) . . .

where the first indicated arrow is the sum of the restrictions, and the second is thedifference.

Proof. The proof is very similar to the proof of theorem 4.2.3, so we will justsketch it. Construct a diagram

0 → Γ(X,G(F)) → Γ(U,G(F))⊕ Γ(V,G(F)) → Γ(U ∩ V,G(F)) → 0↓ ↓ ↓

0 → Γ(X,C1(F)) → Γ(U,C1(F))⊕ Γ(V,C1(F)) → Γ(U ∩ V,C1(F))

and apply the snake lemma to get the sequence of the first 6 terms.

. . . H0(U ∩ V,F)→ H1(X,F) . . .

When applied to C1(F), this yields

. . . H0(U ∩ V,C1(F)) δ→ H2(X,F) . . .

One checks that δ factors through a map

H1(U ∩ V,F) δ→ H2(X,F)

such thatker(δ) = im[H1(U,F)⊕H1(V,F)]

Then repeat with Ci(F) in place of F .

Exercises

1. Complete the proof of theorem 4.5.1.2. Complete the proof of theorem 4.5.2.3. Let A be an Abelian group. Use Mayer-Vietoris to prove that H1(S1, A) ∼=A.

4. Show that H1(Sn, A) = 0 if n ≥ 2.5. Let us say that X has finite dimensional cohomology if

∑dimHi(X,R) <

∞. These dimensions are called the Betti numbers. Then we define thetopological Euler characteristic e(X) =

∑(−1)idimHi(X,R). Show that

if X = U ∪ V and U and V have finite dimensional cohomology, then thesame is true of X, and furthermore e(X) = e(U) + e(V )− e(U ∩ V ).

6. Let S be the topological space associated to a finite simplicial complex(jump ahead to the chapter 7 for the definition if necessary). Prove thate(S) is the alternating sum of the number of simplices.

CHAPTER 5

De Rham cohomology of Manifolds

In this chapter, we study the topology of C∞-manifolds. We define the deRham cohomology of a manifold, which is the vector space of closed differentialforms modulo exact forms. After sheafifying the construction, we see that thede Rham complex forms a so called acyclic resolution of the constant sheaf R.We prove a general result that sheaf cohomology can be computed using suchresolutions, and deduce a version of de Rham’s theorem that de Rham cohomologyis sheaf cohomology with coefficients in R. It follows that de Rham cohomologydepends only on the underlying topology. By using a different acyclic resolutionwhich is dual to the de Rham complex, we prove Poincare duality. This dualitymakes cohomology, which is normally contravariant, into a covariant theory. Wedevote a section to explaining these somewhat mysterious covariant maps calledGysin maps. We end this chapter with the remarkable Lefschetz trace formulawhich, in principle, calculates the number of fixed points for a map of a manifoldto itself.

5.1. Acyclic Resolutions

Definition 5.1.1. A complex of (sheaves of) Abelian groups is a possibly infi-nite sequence

. . . F idi−→ F i+1 di+1−→ . . .

of (sheaves of) groups and homomorphisms satisfying di+1di = 0.

These conditions guarantee that im(di) ⊆ ker(di+1). We denote a complex byF • and we often suppress the indices on d. The ith cohomology of F • is defined by

Hi(F •) =ker(di)

im(di−1).

(We reserve the regular font “H” for sheaf cohomology.) These groups are zeroprecisely when the complex is exact.

Definition 5.1.2. A sheaf F is called acyclic if Hi(X,F) = 0 for all i > 0.

For example, flasque sheaves and soft sheaves on a metric space are acyclic.

Definition 5.1.3. An acyclic resolution of a sheaf F is an exact sequence

0→ F → F0→F1→ . . .

of sheaves such that each F i is acyclic.

A functor between Abelian categories, such as Ab or Ab(X), need not takeexact sequences to exact sequences, but it will always take complexes to complexes.In particular, given a complex of sheaves F•, the sequence

Γ(X,F0)→ Γ(X,F1)→ . . .

69

70 5. DE RHAM COHOMOLOGY OF MANIFOLDS

is necessarily a complex of Abelian groups.

Theorem 5.1.4. Given an acyclic resolution F• of F ,

Hi(X,F) ∼= Hi(Γ(X,F•))

Proof. Let K−1 = F and Ki = ker(F i+1 → F i+2) for i ≥ 0. Then there areexact sequences

0→ Ki−1 → F i → Ki → 0for i ≥ 0. Since each F i is acyclic, Theorem 4.2.3 implies that

(5.1.1) 0→ H0(Ki−1)→ H0(F i)→ H0(Ki)→ H1(Ki−1)→ 0

is exact, and

(5.1.2) Hj(Ki) ∼= Hj+1(Ki−1)

for j > 0. We have a diagram

H0(Ki−1) s

%%LLLLLLLLLL

H0(F i−1) //

88qqqqqqqqqqH0(F i) //

%%JJJJJJJJJH0(F i+1)

H0(Ki)+

99rrrrrrrrrr

which is commutative since the morphism F i−1 → F i factors through Ki−1 and soon. The oblique line in the diagram is part of (5.1.1), so it is exact. In particular,the first hooked arrow is injective. The injectivity of the second hooked arrowfollows for similar reasons. Thus

(5.1.3) im[H0(F i)→ H0(Ki)] = im[H0(F i)→ H0(F i+1)]

Suppose that α ∈ H0(F i) maps to 0 in H0(F i+1), then it maps to 0 in H0(Ki).Therefore α lies in the image of H0(Ki−1). Thus

(5.1.4) H0(Ki−1) = ker[H0(F i)→ H0(F i+1)]

This already implies the theorem when i = 0. Replacing i by i + 1 in (5.1.4), andcombining it with (5.1.1) and (5.1.3) shows that

H1(Ki−1) ∼=H0(Ki)

im[H0(F i)→ H0(Ki)]=

ker[H0(F i+1)→ H0(F i+2)]im[H0(F i)→ H0(F i+1)]

Combining this with the isomorphisms

Hi+1(F) = Hi+1(K−1) ∼= Hi(K0) ∼= . . . H1(Ki−1)

of (5.1.2) proves the theorem for positive exponents.

Example 5.1.5. Let F be a sheaf. Using the notation from section 4.2, defineGi(F) = G(Ci(F)). We define d : Gi(F) → Gi+1(F) as the composition of thenatural maps Gi(F) → Ci+1(F) → Gi+1(F). This can be seen to give an acyclicresolution of F .

Exercises

5.2. DE RHAM’S THEOREM 71

1. Check that G•(F) gives an acyclic resolution of F .2. A sheaf I is called injective if given a monomorphism of sheaves A → B,

any morphism A → I extends to a morphism of B → I. Show that aninjective module is flasque and hence acyclic. (Hint: given an open setU ⊆ X, let ZU = ker[ZX → ZX−U ], check that Hom(ZU ,F) = F(U).)Conclude that if 0 → F → I0 → I1 . . . is an injective resolution, thenHi(X,F) = Hi(Γ(X, I•)); this usually taken as the definition of Hi.

3. A morphism of complexes is a collection of maps F i → Gi commuting withthe differentials d. This would induce a map on cohomology. Suppose thatF → F0 . . . and G → G0 . . . are acyclic resolutions of sheaves F and G, andsuppose that we have a morphism F → G which extends to a morphismof the resolutions. Show that we can choose the isomorphisms so that thediagram

Hi(F) ∼= Hi(Γ(F•))↓ ↓

Hi(G) ∼= Hi(Γ(G•))commutes.

5.2. De Rham’s theorem

Let X be a C∞ manifold and Ek = EkX the sheaf of k-forms on it. Note thatE0X = C∞X . If U ⊂ X is a coordinate neighbourhood with coordinates x1, . . . , xn,Ek(U) is a free C∞(U)-module with basis

dxi1 ∧ . . . ∧ dxik | i1 < . . . < ik

Theorem 5.2.1. There exists canonical R-linear maps d : EkX → Ek+1X , called

exterior derivatives, satisfying the following1. d : E0

X → E1X is the operation introduced in section 2.6.

2. d2 = 0.3. d(α ∧ β) = dα ∧ β + (−1)iα ∧ dβ for all α ∈ E i(X), β ∈ Ej(X).4. If g : Y → X is a C∞ map, g∗ d = d g∗.

Proof. A complete proof can be found in almost any book on manifolds (e.g.[Sk2, Wa]). We will only sketch the construction. When U ⊂ X is a coordinateneighbourhood with coordinates xi, we can see that there is a unique operationsatisfying the above rules given by

d(∑

i1<...<ik

fi1...ikdxi1 ∧ . . . ∧ dxik) =∑

i1<...<ik

∑j

∂fi1...ik∂xj

dxj ∧ dxi1 ∧ . . . dxik

By uniqueness, these local d’s patch.

When X = R3, d can be realized as the div, grad, curl of vector calculus. Thetheorem tells us that E•(X) forms a complex, called the de Rham complex.

Definition 5.2.2. The De Rham cohomology groups (actually vector spaces)of X are defined by

HkdR(X) = Hk(E•(X))

A differential form α is called closed if dα = 0 and exact if it equals dβ forsome β. Elements of de Rham cohomology are equivalence classes [α] representedclosed forms, where two closed forms are equivalent if they differ by an exact form.


Given a C∞ map of manifolds g : Y → X, we get a map g∗ : E∗(X) → E∗(Y ) ofthe de Rham complexes which induces a map g∗ on cohomology. It follows easilythat:

Lemma 5.2.3. X 7→ HidR(X) is a contravariant functor.

We compute the de Rham cohomology of Euclidean space.

Theorem 5.2.4 (Poincare’s lemma). For all n and k > 0,

HkdR(Rn) = 0.

Proof. Assume, by induction, that the theorem holds for n− 1. Consider themaps p : Rn → Rn−1 and ι : Rn−1 → Rn defined by p(x1, x2, . . . xn) = (x2, . . . xn)and ι(x2, . . . xn) = (0, x2, . . . xn). Let I be the identity transformation and letR = (ιp)∗. More explicitly, R : Ek(Rn)→ Ek(Rn) is the R-linear operator definedby

R(f(x1, . . . xn)dxi1∧. . .∧dxik) =

f(0, x2, . . . xn)dxi1 ∧ . . . ∧ dxik if 1 /∈ i1, i2 . . .0 otherwise

where we always choose i1 < i2 < . . .. The image of R can be identified withp∗Ek(Rn−1). Note that R commutes with d. Therefore if α ∈ Ek(Rn) is closed,dRα = Rdα = 0. By the induction assumption, Rα is exact.

For each k, define a linear map h : Ek(Rn)→ Ek−1(Rn) by

h(f(x1, . . . xn)dxi1 ∧ . . . ∧ dxik) =

(∫ x1

0fdx1)dxi2 ∧ . . . ∧ dxik if i1 = 1

0 otherwise

Then the fundamental theorem of calculus shows that dh+ hd = I −R. (In otherwords, h is homotopy from I to R.) Given α ∈ Ek(Rn) satisfying dα = 0. We haveα = dhα+Rα, which is exact.

We have an inclusion of the sheaf of locally constant functions RX ⊂ E0X . This

is precisely the kernel of d : E0X → E1

X .

Theorem 5.2.5. The sequence

0→ RX → E0X → E1

X . . .

is an acyclic resolution of RX .

Proof. Any ball is diffeomorphic to Euclidean space, and any point on amanifold has a fundamental system of such neighbourhoods. Therefore Poincare’slemma implies that the above sequence is exact on stalks, and hence exact.

By corollary 4.4.5, the sheaves Ek are soft hence acyclic.

Corollary 5.2.6 (De Rham’s theorem).

HkdR(X) ∼= Hk(X,R)

Later on, we will work with complex valued differential forms. Essentially thesame argument shows that H∗(X,C) can be computed using such forms.

Exercises

5.3. POINCARE DUALITY 73

1. We will say that a manifold is of finite type if it has a finite open coverUi such that any nonempty intersection of the Ui are diffeomorphic tothe ball. Compact manifolds are known to have finite type [Sk2, pp 595-596]. Using Mayer-Vietoris and De Rham’s theorem, prove that if X is ann-dimensional manifold of finite type, then Hk(X,R) vanishes for k > n,and is finite dimensional otherwise.

2. Let X be a manifold, and let t be the coordinate along R in R×X. Considerthe maps ι : X → R × X and p : R × X → X given by x 7→ (0, x) and(t, x) 7→ x. Since (p ι) = id conclude that ι∗ : Hi

dR(R×X)→ HidR(X) is

surjective.3. Continuing the notation from the previous exercise, let R : Ek(R ×X) →Ek(R×X) be the operator (i p)∗, and h : Ek(R×X)→ Ek−1(R×X) bethe operator which is integration with respect to dt (as in the proof of thePoincare lemma). Show that dh + hd = I − R. Use this to show that Rinduces the identity on Hi

dR(R × X). Conclude that ι∗ : HidR(R × X) →

HidR(X) is also injective, and therefore an isomorphism.

4. Show that C−0 is diffeomorphic to R×S1, and conclude that H1dR(C−

0) is one dimensional. Show that Re(dziz ) = −ydx+xdyx2+y2 generates it.

5. Let Sn denote the n dimensional sphere. Use Mayer-Vietoris with respectto the cover U = Sn−north pole and V = Sn−south pole to computeH∗(Sn,R). (Hint: show that U ∩ V ∼= Sn−1 × R.)

5.3. Poincare duality

Let X be a C∞ manifold. Let Ekc (X) denote the set of C∞ k-forms withcompact support. Since dEkc (X) ⊂ Ek+1

c (X), these form a complex.

Definition 5.3.1. Compactly supported de Rham cohomology is defined byHkcdR(X) = Hk(E•c (X)).

Lemma 5.3.2. For all n,

HkcdR(Rn) =

R if k = n0 otherwise

Proof. [Sk2, chap 8].

This computation suggests that these groups are roughly opposite to the usualde Rham groups. There is another piece of evidence, which is that HcdR behavescovariantly in certain cases. For example, given an open set U ⊂ X, a form in Ekc (U)can be extended by zero to Ekc (X). This induces a map Hk

cdR(U)→ HkcdR(X).

The precise statement of duality requires the notion of orientation. An orienta-tion on an n dimensional real vector space V is a connected component of ∧nV −0(there are two). An ordered basis v1, . . . vn is positively oriented if v1 ∧ . . .∧ vn liesin the given component. If V were to vary, there is no guarantee that we couldchoose a orientations consistently. So we make a definition:

Definition 5.3.3. An n dimensional manifold X is called orientable if ∧nTXminus its zero section has two components. If this is the case, an orientation is achoice of one of these components.


Theorem 5.3.4 (Poincare duality, version I). Let X be a connected orientedn-dimensional manifold. Then

HkcdR(X) ∼= Hn−k(X,R)∗

There is a standard proof of this using currents, which are to forms whatdistributions are to functions. However, we can get by with something much weaker.We define the space of pseudocurrents of degree k on an open set U ⊂ X to be

Ck(U) = En−kc (U)∗ : = Hom(En−kc (U),R).

This is “pseudo” because we are using the ordinary (as opposed to topological) dual.We make this into a presheaf as follows. Given V ⊆ U , α ∈ CkX(U), β ∈ En−kc (V ),define α|V (β) = α(β) where β is the extension of β by 0.

Lemma 5.3.5. CkX is a sheaf.

Proof. Let Ui be an open cover of U , which we may assume is locally finite.Suppose that αi ∈ CkX(Ui) is a collection of sections such that αi|Ui∩Uj = αj |Ui∩Uj .This means that αi(β) = αj(β) if β has support in Ui ∩ Uj . Let ρi be a C∞

partition of unity subordinate to Ui. Then define α ∈ CkX(U) by

α(β) =∑i

αi(ρiβ|Ui)

We have to show that α(β) = αj(β) for any β ∈ En−kc (Uj) with β its extension toU by 0. The support of ρiβ lies in Ui ∩ supp(β) ⊂ Ui ∩Uj , so only finitely many ofthese are nonzero. Therefore

α(β) =∑i

αi(ρiβ) =∑i

αj(ρiβ) = αj(β)

as required. We leave it to the reader to check that α is the unique current withthis property.

Define a map δ : CkX(U) → Ck+1X (U) by δ(α)(β) = (−1)k+1α(dβ). One auto-

matically has δ2 = 0. Thus one has a complex of sheaves.Let X be an oriented n-dimensional manifold. Then we will recall [Sk1] that

one can define an integral∫Xα for any n-form α ∈ Enc (X). Using a partition of

unity, the definition can reduced to the case where α is supported in a coordinateneighbourhood U . Then we can write α = f(x1, . . . xn)dx1 ∧ . . . ∧ dxn, wherethe order of the coordinates are chosen so that ∂/∂x1, . . . , ∂/∂xn gives a positiveorientation of TX . Then∫

X

α =∫

Rn

f(x1, . . . xn)dx1 . . . dxn

The functional∫X

defines a canonical global section of C0X .

Theorem 5.3.6 (Stokes’ theorem). Let X be an oriented n-dimensional man-ifold, then

∫Xdβ = 0.

Proof. See [Sk1].

Corollary 5.3.7.∫X∈ ker[δ].

5.3. POINCARE DUALITY 75

We define a map RX → C0X by sending r → r

∫X

. The key lemma to establishTheorem 5.3.4 is:

Lemma 5.3.8.0→ RX → C0

X → C1X → . . .

is an acyclic resolution.

Proof. Lemma 5.3.2 implies that this complex is exact. Given f ∈ C∞(U)and α ∈ Ck(U) define

fα(β) = α(fβ)

This makes Ck into a C∞-module, and it follows that it is soft and therefore acyclic.

Proof of theorem 5.3.4. We can now use the complex C•X to compute thecohomology of RX to obtain

Hi(X,R) ∼= Hi(C•X(X)) = Hi(En−•c (X)∗).

The right hand space is isomorphic to HicdR(X,R)∗. This completes the proof of

the theorem.

Corollary 5.3.9. If X is a compact oriented n-dimensional manifold. Then

Hk(X,R) ∼= Hn−k(X,R)∗

The following is really a corollary of the proof.

Corollary 5.3.10. If X is a connected oriented n-dimensional manifold. Thenthe map α 7→

∫Xα induces an isomorphism∫

X

: HncdR(X,R) ∼= R

We can make the Poincare duality isomorphism more explicit:

Theorem 5.3.11 (Poincare duality, version II). If f ∈ Hn−kcdR (X)∗, then there

exists a closed form α ∈ Ek(X) such that f([β]) =∫Xα ∧ β. Moreover the class

[α] ∈ HkdR(X) is unique.

Proof. DefineP : EkX → CkX

by

P (α)(β) =∫U

α ∧ β

for α ∈ Ek(U) and β ∈ En−kc (U). With the help of Stokes’ theorem, we see thatδP (α) = P (dα). Therefore, P gives a morphism of complexes of sheaves. Note alsothat P (1) =

∫X

. Thus we have a morphism of resolutions

RX → E•X|| ↓

RX → C•XSo the theorem follows from the exercises of Section 5.1.


If α ∈ E i(X) and β ∈ Ejc (X) are closed forms, then α ∧ β is also closed byTheorem 5.2.1. The cup product of the associated cohomology classes is definedby [α] ∪ [β] = [α ∧ β]. This is a well defined operation which makes de Rhamcohomology into a graded ring when X is compact. The theorem tells us that:

Corollary 5.3.12. The cup product followed by integration gives a nondegen-erate pairing

HkdR(X)×Hn−k

cdR (X)→ HncdR(X) ∼= R

Here is a simple example to illustrate of this.

Example 5.3.13. Consider the torus T = Rn/Zn. We will show later, inSection 8.2, that: Every de Rham cohomology class on T contains a unique formwith constant coefficients. This will imply that there is an algebra isomorphismH∗(T,R) ∼= ∧∗Rn. Poincare duality becomes the standard isomorphism

∧kRn ∼= ∧n−kRn.

Exercises

1. Prove that the Euler characteristic∑

(−1)i dimHi(X) is zero when X isan odd dimensional compact orientable manifold.

2. If X is a connected oriented n dimensional manifold, show that

Hn(X,R) ∼=

R if X is compact0 otherwise

3. (a) Let S2 ⊂ R3 denote unit sphere. Show that

α = xdy ∧ dz + ydz ∧ dx+ zdx ∧ dy

generates H2dR(S2).

(b) The real projective plane RP2 = S2/i, where i(x, y, z) = −(x, y, z).This is a compact manifold. Show that H2

dR(RP2) = 0 by identifyingit with the i∗-invariant part of H2

dR(S2), and conclude that it cannotbe orientable

4. Assuming the exercises of §4.3, prove that Hic(X,R) ∼= Hi

cdR(X).

5.4. Gysin maps

Let f : Y → X be a C∞ map of compact oriented manifolds of dimension mand n respectively. Then we have a natural map

f∗ : HkdR(X)→ Hk

dR(Y )

given by pulling back forms. By Poincare duality, we can identify this with a map

Hn−kdR (X)∗ → Hm−k

dR (Y )∗.

Dualizing and changing variables yields a map in the opposite direction

f! : HkdR(Y )→ Hk+n−m

dR (X),

called the Gysin homomorphism. This is characterized by

(5.4.1)∫X

f!(α) ∪ β =∫Y

α ∪ f∗(β)

5.4. GYSIN MAPS 77

Our goal is to give a more explicit description of this map. Notice that we canfactor f as the inclusion of the graph Y → Y ×X given by y 7→ (y, f(y)), followedby a projection. Therefore we only need to study what happens in these specialcases.

5.4.1. Projections. Suppose that Y = X × Z is a product of compact con-nected oriented manifolds Let p : Y → X and q : Y → Z be the projections. Letr = m − n = dimZ. The fibre integral

∫Z

: Ek(Y ) → Ek−r(X) is defined in localcoordinates by∑

fi1,...ik−n(x1, . . . , xn, z1 . . . zr)dz1 ∧ . . . ∧ dzn ∧ dxi1 ∧ . . . ∧ dxik−r

7→∑(∫fi1,...ik−n

(x1, . . . , xn, z1 . . . zr)dz1, . . . dzn)dxi1 ∧ . . . ∧ dxik−r

Note that∫Zα = 0 if none of its terms contains dz1 ∧ . . . ∧ dzn.

Lemma 5.4.2. p!α is represented by∫Zα.

Proof. One checks that∫Y

α ∧ p∗β =∫X

(∫Z

α) ∧ β

The cohomology of Y has the following structure:

Theorem 5.4.3 (Kunneth formula). The map∑αi ⊗ βj 7→

∑p∗αi ∪ q∗βj

induces an isomorphism⊕i+j=k

HidR(X)⊗R H

jdR(Z) ∼= Hk

dR(Y )

Proof. We will eventually see that de Rham cohomology is the same as sin-gular cohomology. For singular cohomology, a proof can be found in almost anybook on algebraic topology, such as [Hat, Sp].

With respect to the Kunneth decomposition, the Gysin map p! is simply theprojection onto one of the factors

HkdR(Y )→ Hk+n−m

dR (X)⊗Hm−ndR (Z) ∼= Hk+n−m

dR (X)

5.4.4. Inclusions. Now suppose that i : Y → X is an inclusion of a closedsubmanifold. We need the following:

Theorem 5.4.5. There exists an open neighbourhood Tube of Y in X, called atubular neighbourhood. This possesses a C∞ map π : Tube→ Y which makes Tubea locally trivial bundle over Y , with fibres diffeomorphic to Rn−m.

Proof. The details can be found in [Sk2, chap. 9]. However, we give a briefoutline, since we will need to understand a bit about the construction later on. Wechoose a Riemannian metric on X. This amounts to a family of inner products onthe tangent spaces of X; among other things this allows one to define the length ofa curve. The Riemannian distance between two points is the infimum of the lengthsof curves joining the points. This is a metric in the sense of point set topology.


Tube is given by the set of points with Riemannian distance less than ε from Yfor 0 < ε 1. We can take the normal bundle N to be the fibrewise orthogonalcomplement to the tangent bundle TY in TX |Y . N inherits a Riemannian metric,and we let Tube′ ⊂ N be the set of points of distance less than ε from the zerosection. Given a point (y, v) ∈ N , let γy,v(t) be the geodesic emanating from y withvelocity v. Then the map (y, v) 7→ γy,v(1) defines a diffeomorphism from Tube′ toTube.

Then the map i∗ can be factored as

Hm−kdR (X)→ Hm−k

dR (Tube) ∼→ Hm−kdR (Y )

The second map is an isomorphism, since the fibres of π are contractible. Dualizing,we see that i! is a composition of

HkdR(Y ) ∼→ Hk+n−m

cdR (Tube)→ Hk+n−mdR (X)

The first map is called the Thom isomorphism. The second map can be seen tobe extension by zero. To get more insight into this, let k = 0. Then H0

dR(Y ) hasa natural generator, which is the constant function 1Y with value 1. Under theThom isomorphism, this maps to a class τY ∈ Hn−m

cdR (T ), called the Thom class.This can be represented by (any) differential form with compact support in Tube,which integrates to 1 on the fibres of π. The Thom isomorphism is given explicitlyby α 7→ π∗α ∪ τY . So to summarize,

Lemma 5.4.6. i!α is the extension of π∗α ∪ τY to X by zero.

Exercises

1. Let j : U → X be the inclusion of an open set in a connected ori-ented manifold. Check that the Poincare dual of the restriction mapj∗ : H∗dR(X)→ H∗dR(U) is given by extension by zero.

2. Fill in all the details for the proof of lemma 5.4.6.3. Prove the projection formula f!(f∗(α) ∪ β) = α ∪ f!β

5.5. Fundamental class

We can use Gysin maps to construct interesting cohomology classes. Let Y ⊂ Xbe a closed connected oriented m dimensional submanifold of an n dimensionaloriented manifold. Denote the inclusion by i, as before.

Definition 5.5.1. The fundamental class of Y in X is [Y ] = i!1Y ∈ Hn−mdR (X).

Equivalently, [Y ] is the extension of τY by zero. Under the duality isomorphism,H0dR(Y ) ∼= Hm

dR(Y )∗, 1 goes to the functional

β 7→∫Y

β,

and this maps to

α 7→∫Y

i∗α

in Hm(X)∗. Composing this with the isomorphism Hm(X)∗ ∼= Hn−m(X), yieldsthe basic relation

(5.5.1)∫Y

i∗α =∫X

[Y ] ∪ α

5.5. FUNDAMENTAL CLASS 79

Let Y,Z ⊂ X be oriented submanifolds such that dimY + dimZ = n. Thenunder the duality isomorphism, [Y ]∪ [Z] ∈ Hn(X,R) ∼= R corresponds to a numberY · Z, called the intersection number. This has a geometric interpretation whichwe give under an extra assumption that holds “most of the time”. We say that Yand Z are transverse if Y ∩ Z is finite and if TY,p ⊕ TZ,p = TX,p for each p in theintersection.

Definition 5.5.2. Let Y and Z be transverse, and let p ∈ Y ∩ Z. Choose or-dered bases v1(p), . . . vm(p) ∈ TY,p and vm+1(p) . . . vn(p) ∈ TZ,p which are positivelyoriented with respect to the orientations of Y and Z. The local intersection numberat p is

ip(Y,Z) =

+1 if v1(p) . . . vm(p), vm+1(p), . . . vn(p) is a positively oriented basis of TX,p−1 otherwise

(This is easily seen to be independent of the choice of bases. )

Proposition 5.5.3. If Y and Z are transverse, Y · Z =∑p∈Y ∩Z ip(Y, Z).

Proof. Let m = dimY . Let Up be a collection of disjoint coordinate neigh-bourhoods for each p ∈ Y ∩ Z. (Note that these Up will be replaced by smallerneighbourhoods whenever necessary.) Choose coordinates x1, . . . xn around each pso that Y is given by xm+1 = . . . xn = 0 and Z by x1 = . . . xm = 0. Next constructsuitable tubular neighbourhoods π : T → Y of Y and π′ : T ′ → Z of Z. Recallthat these neighbourhoods depend on a choice of Riemannian metric and radii ε, ε′.By choosing the radii small enough, we can guarantee that T ∩ T ′ lies in the unionof Up. Also, by modifying the metric to be Euclidean near each p, we can assumethat π is locally the projection (x1 . . . xn) 7→ (x1, . . . xm), and likewise for π′.

Then with the above assumptions

Y · Z =∫X

τY ∧ τZ =∑p

∫Up

τY ∧ τZ

where τY and τZ are forms representing the Thom classes of T and T ′. We willassume that Up is a ball and hence diffeomorphic to Rn. We can view τY |Up

asdefining a class in

H0dR(Rm)⊗Hn−m

cdR (Rn−m) ∼= Hn−mcdR (Rn−m)

and similarly for τZ |Up. Thus we can see that

τY |Up= f(xm+1, . . . xn)dxm+1 ∧ . . . ∧ dxn + dη

τZ |Up = g(x1, . . . xm)dx1 ∧ . . . ∧ dxm + dξ

with f and g compactly supported such that∫Rn−m

f(xm+1, . . . xn)dxm+1 . . . dxn =∫

Rm

g(x1, . . . xm)dx1 . . . dxm = 1.

Fubini’s theorem and Stokes’ theorem then gives∫Up

τY ∧ τZ = ip(Y,Z)


The proposition implies that the intersection number is an integer for transverseintersections. In fact, this is always true. There a couple of ways to see this. One isby proving that intersections can always be made transverse. A simpler explanationis that the fundamental classes can actually be defined to take values in integralcohomology H∗(X,Z). Moreover, we have a cup product pairing

Hk(X,Z)×Hn−k(X,Z)→ Hn(X,Z) ∼= Z

The classes that we have defined are images under the natural map H∗(X,Z) →H∗(X,R).

Example 5.5.4. Let T = Rn/Zn, and let ei be the standard basis, and let xibe coordinates on Rn. If VI ⊂ Rn is the span of ei | i ∈ I, then TI = VI/(Zn∩VI)is a submanifold of T . Its fundamental class is dxi1 ∧ . . .∧dxid , where i1 < . . . < idare the elements of I in increasing order. If J is the complement of I, then TI andTJ will meet transversally at one point. Therefore TI · TJ = ±1.

Example 5.5.5. Consider complex projective space PnC. This is the basic ex-ample for us, and it will be studied further in Section 7.2. For now we just statethe main results. We have

HidR(Pn) =

R if 0 ≤ i ≤ 2n is even0 otherwise

Given a complex subspace V ⊂ Cn+1, the subset P(V ) ⊂ Pn consisting of lines lyingin V forms a submanifold, which can be identified with another projective space. IfW is another subspace with dimW = n − dimV + 2 and dim(V ∩W ) = 1, thenP(V ) and P(W ) will meet transversally at one point. In this case P(V ) · P(W ) isnecessarily +1 (see the exercises).

Exercises

1. Show that if Y,Z ⊂ X are transverse complex submanifolds of a complexmanifold, then ip(Y,Z) = 1 for each p in the intersection. Thus Y · Z isthe number of points of intersection.

2. Check that fundamental classes of subtori of Rn/Zn are described as above.3. Let T = R2/Z2 and let V,W ⊂ R2 be distinct lines with rational slope.

Show that the images of V and W in T are transverse. Find an interpre-tation for their intersection number.

5.6. Lefschetz trace formula

Let X be a compact n dimensional oriented manifold with a C∞ map f : X →X. The Lefschetz formula is a formula for the number of fixed points countedappropriately. This needs to be explained. Let

Γf = (x, f(x)) | x ∈ X

∆ = (x, x) | x ∈ Xbe the graph of f and the diagonal respectively. These are both n dimensionalsubmanifolds of X×X which intersect precisely at points (x, x) with x = f(x). Wedefine the “number of fixed points” as Γf ·∆. Since this number could be negative,

5.6. LEFSCHETZ TRACE FORMULA 81

we need to take this with grain of salt. If these manifolds are transverse, we seethat this can be evaluated as the sum over fixed points∑

x

ix(Γf ,∆)

Theorem 5.6.1. The number Γf ·∆ is given by

L(f) =∑p

(−1)p trace[f∗ : Hp(X,R)→ Hp(X,R)]

Proof. The proof will be based on the elementary observation that if F is anendomorphism of a finite dimensional vector space with basis vi and dual basisv∗i , then

trace(F ) =∑i

v∗i (F (vi))

For each p, choose a basis αp,i of Hp(X), and let α∗p,i denote the dual basistransported to Hn−p(X) under the Poincare duality isomorphism Hn−p(X) ∼=Hp(X)∗. Let πi : X × X → X denote the projections. Then by Kunneth’s for-mula Ap,i,j = π∗1αp,i ∪ π∗2α∗p,jp,i,j gives a basis for Hn(X × X), and A∗p,i,j =(−1)n−pπ∗1α

∗p,i ∪ π∗2αp,j is seen to give a basis which is dual to this in the sense

that ∫X×X

Ap,i,j ∪A∗p′,i′,j′ = δ(p,i,j),(p′,i′,j′)

Thus we can express[∆] =

∑cp,i,jAp,i,j

The coefficients can be computed by integrating against the dual basis

cp,i,j =∫X×X

[∆] ∪A∗p,i,j =∫

∆

A∗p,i,j = (−1)n−p∫X

αp,i ∪ αp,j = (−1)pδij

Therefore

(5.6.1) [∆] =∑i,p

(−1)n−pπ∗1αp,i ∪ π∗2α∗p,i

Consequently,

Γf ·∆ =∫

Γf

[∆]

=∑p

(−1)n−p∑i

∫Γf

π∗1αp,i ∪ π∗2α∗p,i

=∑p

(−1)n−p∑i

∫X

αp,i ∪ f∗α∗p,i

=∑p

(−1)n−p trace[f∗ : Hn−p(X,R)→ Hn−p(X,R)]

= L(f)

Corollary 5.6.2. If L(f) 6= 0, then f has a fixed point.

Proof. If Γf ∩∆ = ∅, then Γf ·∆ = 0.


Corollary 5.6.3. ∆ ·∆ is the Euler characteristic e(X)

Exercises

1. We say that two C∞ maps f, g : X → Y between manifolds are homotopicif there exists a C∞ map h : X × R → Y such that f(x) = h(x, 0) andg(x) = h(x, 1). Using the exercises of Section 5.2, show that f∗ = g∗ if fand g are homotopic. Conclude that g has a fixed point if L(f) 6= 0.

2. Let v(x) be C∞ vector field on a compact manifold X. By the existence anduniqueness theorem for ordinary differential equations, there is an ε > 0such that for each x there is unique curve γx : [0, ε] → X with γx(0) = xand dγx(t) = v(γx(t)). Moreover, the map x 7→ γx(δ) is a diffeomorphismfrom X to itself for every δ ≤ ε. Use this to show that v must have a zeroif e(X) 6= 0.

3. Let A be a nonsingular n× n matrix. Then it acts on Pn−1 by [v] 7→ [Av],and the fixed points correspond to eigenvectors. Show that A is homotopicto the identity. Use this to show that L(A) 6= 0, and therefore that A hasan eigenvector. Deduce the fundamental theorem of algebra from this.

CHAPTER 6

Riemann Surfaces

Riemann surfaces are the same thing as one dimensional complex manifoldsor nonsingular complex curves. We already considered these briefly in the firstchapter. But now we are in a position to make a more thorough study by using thegeometric and homological tools introduced in the intervening chapters.

6.1. Topological Classification

A Riemann surface can be regarded as a manifold of real dimension 2. It hasa canonical orientation: if we identify the real tangent space at any point withthe complex tangent space, then for any nonzero vector v, we declare the orderedbasis (v, iv) to be positively oriented. Let us now forget the complex structureand consider the purely topological problem of classifying these surfaces up tohomeomorphism.

Given two connected 2 dimensional topological manifolds X and Y with pointsx ∈ X and y ∈ Y , we can form new topological manifold X#Y called the connectedsum. To construct this, choose open disks D1 ⊂ X and D2 ⊂ Y . Then X#Y isobtained by gluing X−D1∪S1×[0, 1]∪Y −D2 appropriately. The homeomorphismclass is independent of the choices made. Figure (1) depicts the connected sum oftwo tori.

a

b1

a21

b2

Figure 1. Genus 2 Surface

Theorem 6.1.1. A compact connected orientable 2 dimensional topologicalmanifold is classified, up to homeomorphism, by a nonnegative integer called thegenus. A genus 0 surface is homeomorphic to the 2-sphere S2. A manifold of genusg > 0 is homeomorphic to a connected sum of the 2-torus and a surface of genusg − 1.

Proof. An equivalent formulation is that every such manifold is homeomor-phic to a sphere with g handles. A classical reference for this is [ST, p. 145].

There is another standard model for these surfaces [ST] which is also quiteuseful (for instance for computing the fundamental group). A genus g surface can

83

84 6. RIEMANN SURFACES

be constructed by gluing the sides of a 2g-gon. It is probably easier to visualizethis in reverse. After cutting the genus 2 surface of (1) along the indicated curves,it can be opened up to an octagon (2).

a1 b1

a1

b1

b2

b2

a2

a2

Figure 2. Genus 2 surface cut open

The topological Euler characteristic of space X is

e(X) =∑

(−1)i dim Hi(X,R).

From the exercises of Section 4.5, we have:

Lemma 6.1.2. If X is a union of two open sets U and V , then e(X) = e(U) +e(V )− e(U ∩ V ).

Corollary 6.1.3. If X is a manifold of genus g, then e(X) = 2− 2g and thefirst Betti number dimH1(X,R) = 2g.

Proof. Exercise.

When g = 2, this gives dimH1(X,R) = 4. We can find explicit generators bytaking the fundamental classes of the curves a1, a2, b1, b2 in figure (1), after choosingorientations. To see that these generate, it suffices to prove that they are linearlyindependent. For this, consider the pairing

(α, β) 7→∫X

α ∧ β

on H1(X,C). This restricts to the intersection pairing on fundamental classes.After orienting the curves suitably, their intersection matrix is

0 0 1 00 0 0 1−1 0 0 00 −1 0 0

and this shows independence. A similar basis with intersection matrix(

0 I−I 0

)can be found for any g. These basis vectors will generate a lattice inside H1(X,R)which can be identified with H1(X,Z).

Exercises

6.2. EXAMPLES 85

1. Prove corollary 6.1.3.

6.2. Examples

Many examples of compact Riemann surfaces can be constructed explicitlyas nonsingular projective algebraic curves. (Later, we will see that all compactRiemann surfaces arise this way.)

Example 6.2.1. Let f(x, y, z) be a homogeneous polynomial of degree d. Sup-pose that the partials of f have no common zeros in C3 except (0, 0, 0). Then thecurve V (f) = f(x, y, z) = 0 in P2

C is nonsingular. We will see later that thegenus is (d−1)(d−2)

2 . In particular, not every genus occurs for these examples.

Example 6.2.2. Given a collection of homogeneous polynomials fi ∈ C[x0, . . . xn]such that X = V (f1, f2 . . .) ⊂ PnC is a nonsingular algebraic curve. Then X will acomplex submanifold of PnC and hence a Riemann surface. By a generic projectionargument, n = 3 suffices to give all such examples.

Example 6.2.3. Choose 2g + 2 distinct points in ai ∈ C. Consider the affinecurve C1 ⊂ C2 defined by

y2 =∏

(x− ai)We can compactify this, to obtain a nonsingular projective curve C this by takingthe projective closure and normalizing. Here is an alternative description of thesame curve. Consider the affine curve C2:

Y 2 =∏

(1− aiX)

Glue this to C1 by identifying X = x−1 and Y = yx−g−1. Since C is nonsingular,it can be viewed as a Riemann surface. By construction, C comes equipped witha morphism f : C → P1

C which is 2 to 1 except at the branch points ai. In theexercises, it will be shown that the genus of X is g. Curves of this form are calledhyperelliptic (if g > 1), and very nice to work with. However, “most” curves arenot hyperelliptic.

From a more analytic point of view, we can construct many examples as quo-tients of C or the upper half plane. In fact, the uniformization theorem tells us thatall examples other than P1 arise this way.

Example 6.2.4. Let L ⊂ C be a lattice, i. e. an Abelian subgroup generatedby two R-linearly independent numbers. The quotient E = C/L can be made intoa Riemann surface (exercises from 2.2) called an elliptic curve. Since this topo-logically a torus, the genus is 1. Conversely, any genus 1 curve is of this form.

E = C/L can be realized as an algebraic curve in a couple of ways. As we sawin the first chapter, we can embed it into P2 as the cubic curve

(6.2.1) zy2 = 4x3 − g2(L)xz2 − g3(L)z3

by z 7→ [℘(z), ℘′(z), 1]. Alternatively, ℘ : E → P1 realizes this directly as a twosheeted cover branched at 4 points. We can we can assume without loss of generalitythat these are 0, 1,∞, t. Then E is given by Legendre’s equation

(6.2.2) y2z = x(x− z)(x− tz)


Let us now consider quotients of the upper half plane H = z | Im(z) > 0.The group SL2(R) acts on H by fractional linear transformations:

z 7→ az + b

cz + d

Note that−I acts trivially, so the action factors through PSL2(R) = SL2(R)/±I.The action of subgroup Γ ⊂ SL2(R) on H is properly discontinuous if every pointhas a neighbourhood V such that γV ∩ V 6= ∅ for all but finitely many γ ∈ Γ. Apoint x ∈ V is called a fixed point if its stabilizer in Γ/±I is nontrivial. Theaction is free if it has no fixed points.

Proposition 6.2.5. If Γ acts properly discontinuously on H, the quotient X =H/Γ becomes a Riemann surface. If π : H → X denotes the projection, define thestructure sheaf f ∈ OX(U) if and only if f π ∈ OH(π−1U).

When Γ acts freely with compact quotient, then it has genus g > 1. Thequickest way to see this is by applying the Gauss-Bonnet theorem, which says thate(H/Γ) = 2 − 2g can be computed by integrating the Gaussian curvature for anyRiemannian metric. The hyperbolic metric on H, which descends to the quotient,has negative curvature. A fundamental domain for this action is a region R ⊂ Hsuch that ∪γγR = H and such that two translates of R can only meet at theirboundaries. The fundamental domain in this case can be chosen to be the interiorof a geodesic 2g-gon.

The modular group, SL2(Z), is a particularly important example, where theaction is not free.

Theorem 6.2.6. The quotient H/SL2(Z) is isomorphic to C, and its pointscorrespond to isomorphism classes elliptic curves (which are genus one curves witha distinguished point).

Proof. We sketch the idea. The details can be found in [DS, Si2] or [S4].A fundamental domain for the action is given in figure 3. The marked pointsρ = e2πi/3, i, ρ+ 1.

T

S

fixed pts

Figure 3. fundamental domain of SL2(Z)

6.2. EXAMPLES 87

After identifying the sides of the domain by the transformations

S =(

0 −11 0

), T =

(1 10 1

)as indicated in the diagram, one sees that that the quotient H/SL2(Z) is home-omorphic to C. By the Riemann mapping theorem, H/SL2(Z) isomorphic to ei-ther C or H. To see that it is C, we show that the one point compactificationH/SL2(Z) ∪ ∗ can also be made into a Riemann surface, necessarily isomorphicto P1. Any continuous function f defined in a neighbourhood of ∗ can be pulledback to an invariant function f on H. Invariance under T implies that f is periodic,hence it can expanded in a Fourier series

f(z) =∞∑

n=−∞ane

2πinz

We declare f to be holomorphic if all the an = 0 for n < 0. To put this anotherway, q = e2πiz is set as a local analytic coordinate at ∗.

The relation to elliptic curves is given in the exercises.

Exercises

1. Check that the genus of the hyperelliptic curve constructed above is g bytriangulating in such way that the ai are included in the set of vertices.

2. Let g(x, y, z) be a homogeneous polynomial of degree d, such that its par-tials have no common zeros. Let f0(x, y, z) be a product of d linear poly-nomials, and let ft(x, y, z) = f0 + tg. The variety V (f0) is a union of linesLi, and let X = V (ft) for small t. Choose small closed balls Bij aroundeach crossing point. Show that Bij ∩ X is diffeomorphic to the cylinderS1 × [0, 1].

3. Continuing the notation from the previous exercise, the complements X −∪Bij and V (f0) − ∪Bij can be seen to be diffeomorphic using a versionof Ehressman’s Theorem 13.1.1. Granting this, we obtain the followingtopological model for X. Take d copies L1, . . . Ld of S2, remove d−1 disjointdisks Di,j from each Li, and join Li too Lj by gluing a cylinder from ∂Di,j

to ∂Dj,i to get X. Use this to verify that its genus is (d− 1)(d− 2)/2.4. Fix τ, τ ′ ∈ H. Check that the following are equivalent:

(a) The elliptic curves C/Z + Zτ and C/Z + Zτ ′ are isomorphic i.e. thereexists a holomorphic bijection preserving 0 and taking Z + Zτ to Z +Zτ ′.

(b) There exists λ ∈ C∗ such that λ(Z + Zτ) = Z + Zτ ′.(c)

τ ′ =aτ + b

cτ + d

with(a bc d

)∈ SL2(Z).


6.3. The ∂-Poincare lemma

Let U ⊂ C be an open set. Let x and y be real coordinates on C, and z = x+iy.Given a complex valued C∞ function f : U → C, let

∂f

∂z=

12

(∂f

∂x− i∂f

∂y)

∂f

∂z=

12

(∂f

∂x+ i

∂f

∂y).

With this notation, the Cauchy-Riemann equation is simply ∂f∂z = 0. Define the

complex valued 1-forms dz = dx+ idy and dz = dx− idy. With this notation, wecan formulate Cauchy’s formula for C∞ functions.

Theorem 6.3.1. Let D ⊂ C be a disk. If f ∈ C∞(D), then

f(ζ) =1

2πi

∫∂D

f(z)z − ζ

dz +1

2πi

∫D

∂f(z)∂z

dz ∧ dzz − ζ

Proof. This follows from Stokes’ theorem, see [GH, pp. 2-3].

The following as analogue of the Poincare lemma.

Theorem 6.3.2. Let D ⊂ C be an open disk. Given f ∈ C∞(D), the functiong ∈ C∞(D) given by

g(ζ) =1

2πi

∫D

f(z)z − ζ

dz ∧ dz

satisfies ∂g∂z = f .

Proof. Decompose f(z) = f1(z) + f2(z) into a sum of C∞ functions, wheref1(z) ≡ f(z) in a small neighbourhood of z0 ∈ D and vanishes near the boundaryof D. In particular, f2 is zero in neighbourhood of z0. Let g1 and g2 be thefunctions obtained by substituting f1 and f2 for f in the integral of the theorem.Differentiating under the integral sign yields

∂g2(ζ)∂ζ

=1

2πi

∫D

∂

∂ζ

(f2(z)z − ζ

)dz ∧ dz =

12πi

∫D

d

(f2(z)dzz − ζ

)= 0

for ζ close to z0. Since, f1 is compactly supported,

g1(ζ) =1

2πi

∫C

f1(z)z − ζ

dz ∧ dz

Then doing a change of variables w = z − ζ,

g1(ζ) =1

2πi

∫C

f1(w + ζ)w

dw ∧ dw

Thus for ζ close to z0

∂g(ζ)∂ζ

=∂g1(ζ)∂ζ

=1

2πi

∫C

∂f1(w + ζ)∂ζ

dw ∧ dww

=1

2πi

∫D

∂f1(z)∂z

dz ∧ dzz − ζ

Since f1 vanishes on the boundary, the last integral equals f1(ζ) = f(ζ) byCauchy’s formula 6.3.1.

6.4. ∂-COHOMOLOGY 89

In order to make it easier to globalize the above operators to Riemann surfaces,we reinterpret them in terms of differential forms. Let C∞(U) and E•(U) will denotethe space of complex valued C∞ functions and forms on U ⊆ C (we will continuethis convention from now on). The exterior derivative extends to a C-linear operatorbetween these spaces. Set

∂f =∂f

∂zdz

∂f =∂f

∂zdz

We extend this to 1-forms, by

∂(fdz) =∂f

∂zdz ∧ dz

∂(fdz) = 0

∂(fdz) =∂f

∂zdz ∧ dz

∂(fdz) = 0

A 1-form α is holomorphic α = fdz with f holomorphic. This is equivalent tothe condition ∂α = 0. The following identities can be easily verified:

d = ∂ + ∂(6.3.1)∂2 = ∂2 = 0∂∂ + ∂∂ = 0

Exercises

1. Check the identities (6.3.1).2. Show that a form fdz is closed if and only if it is holomorphic.

6.4. ∂-cohomology

Let X be a Riemann surface manifold with OX and Ω1X its sheaf of holomor-

phic functions and 1-forms respectively. We write C∞X and EnX for the sheaves ofcomplex valued C∞ functions and n-forms. We define a C∞X -submodule E(1,0)

X ⊂ E1X

(respectively E(0,1)X ⊂ E1

X), by E(1,0)X (U) = C∞(U)dz (resp. E(0,1)

X (U) = C∞(U)dz)for any coordinate neighbourhood U with holomorphic coordinate z. We have adecomposition

E1X = E(1,0)

X ⊕ E(0,1)X

We set E(1,1)X = E2

X as this is locally generated by dz ∧ dz.

Lemma 6.4.1. There exists C-linear maps ∂, ∂ on the sheaves E•X which coincidewith the previous expressions in local coordinates.

It follows that the identities (6.3.1) hold globally. We have inclusions OX ⊂ C∞Xand Ω1

X ⊂ E(1,0)X .

Lemma 6.4.2. The sequences of sheaves

0→ OX → C∞X∂−→ E(0,1)

X → 0

0→ Ω1X → E

(1,0)X

∂−→ E(1,1)X → 0


are acyclic resolutions.

Proof. The exactness can be checked on a disk where it follows from Theorem6.3.2. The sheaves C∞X , E•X are C∞-modules and hence soft.

Corollary 6.4.3.

H1(X,OX) =E(0,1)(X)∂C∞(X)

H1(X,Ω1X) =

E(1,1)(X)

∂E(1,0)X (X)

andHi(X,OX) = Hi(X,Ω1

X) = 0if i > 1.

Next, we give a holomorphic analogue of the de Rham complex.

Proposition 6.4.4. There is an exact sequence of sheaves

0→ CX → OXd−→ Ω1

X → 0

Proof. The only nontrivial part of the assertion is that OX → Ω1X is an

epimorphism. We can check this by replacing X by a disk D. A holomorphic 1-form α on D is automatically closed, therefore α = df by the usual Poincare lemma.Since df is holomorphic, ∂f = 0. Therefore f is holomorphic.

Corollary 6.4.5. There is a long exact sequence

0→ H0(X,C)→ H0(X,OX)→ H0(X,Ω1X)→ H1(X,C) . . .

Holomorphic 1-forms are closed, and

H0(X,Ω1X)→ H1(X,C)

is the map which sends a holomorphic form to its class in (complex valued) deRham cohomology.

Lemma 6.4.6. When X is compact and connected, this map is an injection.

Proof. Since global holomorphic functions on X are constant (proposition2.7.2),

H0(X,C)→ H0(X,OX)is surjective.

We will postpone the following proposition to section 9.1.

Proposition 6.4.7. The dimensions of H0(X,Ω1X) and H1(X,OX) both co-

incide with the genus.

Corollary 6.4.8. H1(X,Ω1X) ∼= C.

Proof. The proposition implies that the map

H1(X,C)→ H1(X,OX)

is surjective, and therefore that H1(X,Ω1X) is isomorphic to H2(X,C) = C.

Exercises

6.5. PROJECTIVE EMBEDDINGS 91

1. Let V be a vector space with a nondegenerate skew symmetric pairing 〈, 〉.A subspace W ⊂ V is called isotropic if 〈w,w′〉 = 0 for all w,w′ ∈ W .Prove that dimW ≤ dimV/2 if W is isotropic.

2. Show that H0(X,Ω1X) is isotropic for the pairing

∫α ∧ β. Use this to

conclude that dim H0(X,Ω1X) ≤ g.

3. Show that the differentials xidx/y, with 0 ≤ i < g, are holomorphic on thehyperelliptic curve 6.2.3. Conclude directly that H0(X,Ω1

X) = g in thiscase.

6.5. Projective embeddings

Fix a compact Riemann surface X. We introduce some standard shorthand:hi = dim Hi, and ωX = Ω1

X . We will follow standard practice of referring to locallyfree OX -modules of rank one as “line bundles”. (As we will see there is a one toone correspondence, so this not unreasonable.)

Definition 6.5.1. A divisor D on X is a finite integer linear combination∑nipi where pi ∈ X. It is effective if all the coefficients are nonnegative. The

degree deg D =∑ni.

Divisors form an Abelian group Div(X) in the obvious way. For every mero-morphic function defined in a neighbourhood of p ∈ X, let ordp(f) be the order ofvanishing (or minus the order of the pole) of f at p. If D is a divisor define ordp(D)to be the coefficient of p in D (or 0 if p is absent). Define the sheaf OX(D) by

OX(D)(U) = f : U → C ∪ ∞ meromorphic | ordp(f) + ordp(D) ≥ 0, ∀p ∈ U

Lemma 6.5.2.1. OX(D) is a line bundle.2. OX(D +D′) ∼= OX(D)⊗OX(D′).

Proof. Let z be a local coordinate defined in a neighbourhood U . Let D =∑nipi+E where pi ∈ U and E is a sum of points not in U . Then it can be checked

thatOX(D)(U) = OX(U)

1(z − p1)n1(z − p2)n2 . . .

,

and this is free of rank one. If D′ =∑n′ipi + E′ is a second divisor.

[O(D)⊗O(D′)](U) ∼= OX(U)1

(z − p1)n1 . . .

1(z − p1)n′1 . . .

= O(D +D′)(U)

In later terminology, this says that D 7→ OX(D) is a homomorphism fromDiv(X) → Pic(X). If D is effective O(−D) is a sheaf of ideals. In particular,OX(−p) is exactly the maximal ideal sheaf at p. We have an exact sequence

(6.5.1) 0→ OX(−p)→ OX → Cp → 0

where

Cp(U) =

C if p ∈ U0 otherwise

is a so called skyscraper sheaf. Observe that Cp ⊗ L ∼= Cp for any line bundle L.Therefore tensoring (6.5.1) by OX(D) yields

(6.5.2) 0→ OX(D − p)→ OX(D)→ Cp → 0


In the same way, we get a sequence

(6.5.3) 0→ ωX(D − p)→ ωX(D)→ Cp → 0

where ωX(D) = ωX ⊗OX(D).

Lemma 6.5.3. For all D, Hi(X,OX(D)) and Hi(X,ωX(D)) are finite dimen-sional and 0 if i > 1.

Proof. We prove this for OX(D), since the argument for ωX(D) is the same.Let D =

∑npp. The proof goes by induction on d =

∑|np|. The initial case D = 0

follows from corollary 6.4.3.Observe that Cp is flasque, and thus has no higher cohomology. (6.5.2) yields

0→ H0(OX(D − p))→ H0(OX(D))→ C→ H1(OX(D − p))→ H1(OX(D))→ 0

and isomorphismsHi(OX(D − p)) ∼= Hi(OX(D))

for i > 1 By adding or subtracting a point, we can drop d by one, and use induction.

A meromorphic 1-form is a holomorphic form on the complement of a finite setS such that it has finite order poles at the points of S. The residue of a meromorphic1-form α at p is

resp(α) =1

2πi

∫C

α

where C is any loop “going once counterclockwise” around p and containing nosingularities other than p. Alternatively, if α = f(z)dz locally for some local coor-dinate z at p, resp(α) is the coefficient of 1

z in the Laurent expansion of f(z).

Lemma 6.5.4 (Residue Theorem). If α is a meromorphic 1-form, the sum ofits residues is 0.

Proof. p1, . . . pn. For each i, choose an open disk Di containing pi and noother singularity. Then by Stokes’ theorem∑

respiα =1

2πi

∫X−∪Di

dα = 0.

Theorem 6.5.5. Suppose that D is a nonzero effective divisor then(a) (Kodaira vanishing) H1(ωX(D)) = 0.(b) (Weak Riemann-Roch) h0(ωX(D)) = deg D + g − 1.

Proof. This can be proved by induction on the degree of D. We carry out theinitial step here; the rest is left as an exercise. Suppose D = p. Then H0(ωX(p))consists of the space of meromorphic 1-forms α with at worst a simple pole atp and no other singularities. The residue theorem implies that such an α mustbe holomorphic. Therefore H0(ωX(p)) = H0(ωX). Therefore h0(ωX(p)) = g aspredicted by (b). By the long exact sequence of cohomology groups associated to(6.5.3), we have

0→ C→ H1(ωX)→ H1(ωX(p))→ 0Since the space in the middle is one dimensional by corollary 6.4.8, H1(ωX(p)) = 0.

6.5. PROJECTIVE EMBEDDINGS 93

Corollary 6.5.6. There exists a divisor K (called a canonical divisor) suchthat ωX ∼= OX(K). If g > 0, then K can be chosen to be effective.

Proof. Choose D so that H0(ωX(D)) possesses a nonzero section α. Locallyα = fdz, and we define ordp(α) = ordp(f) (this is independent of the coordinatez). Then

K = (α)−D =∑

ordp(α)p−Dsatisfies the required properties.

The degree of K is known.

Proposition 6.5.7. For any canonical divisor, deg K = 2g − 2

We say that a line bundle L on X is globally generated if for any point x ∈ X,there exists a section f ∈ H0(X,L) such that f(x) 6= 0. Suppose that this is thecase, Choose a basis f0, . . . fN for H0(X,L). If we fix an isomorphism τ : L|U ∼= OU ,τ(fi) are holomorphic functions on U . Thus we get a holomorphic map U → CN+1

given by x 7→ (τ(fi(x))). By our assumption, the image lies in the complement of0, and thus descends to a map to projective space. The image is independent of τ ,hence we get a well defined holomorphic map

φL : X → PN

This map has the property that φ∗LOPN (1) = L. L is called very ample if φL is aclosed immersion, that is an isomorphism onto φL(X).

Proposition 6.5.8. A sufficient condition for L to be globally generated is thatH1(X,L(−p)) = 0 for all p ∈ X A sufficient condition for L to be very ample isthat H1(X,L(−p− q)) = 0 for all p, q ∈ X

Proof. We have an exact sequence

0→ L(−p)→ L → Cp → 0

Then H1(L(−p)) = 0, implies that H0(L) surjects onto H0(Cp). This implies thatL has a global section which is nonzero at p.

To show that φL(X) is a closed immersion, we need to check that it is injectiveand injective on tangent spaces. When p 6= q, the assumption H1(L(−p− q)) = 0would imply that H0(L) surjects onto H0(Cp ⊕ Cq). Therefore L has a sectionsvanishing at p but not q and at q but not p. This shows that φL is injective. Ifp = q, then see that H0(L) surjects onto H0(Op/m2

p). Therefore there is a sectionfi with nonzero image in mp/m

2p = T ∗p

∼= C. After identifying fi with a function,this just means f ′i(p) 6= 0. Thus the derivative of φL at p is nonzero, and henceinjective.

Corollary 6.5.9. ωX(D) is very ample if D is nonzero effective with degD >2. In particular, any compact Riemann surface can be embedded into a projectivespace.

Using Chow’s Theorem 15.5.5, we obtain

Corollary 6.5.10. Every compact Riemann surface is isomorphic to an non-singular projective algebraic curve.

Exercises


1. Finish the proof of theorem 6.5.5, by writing D = p + D′ and applying(6.5.3).

2. A divisor is called principal, if it is given by (f) =∑ordp(f)p for some

nonzero meromorphic function f . Prove that a principle divisor has degree0. Conclude that any two canonical divisors have the same degree.

3. Prove proposition 6.5.7 by checking it for P1, and then showing that behaveslike Riemann-Hurwitz for a branched covering f : X → P1

4. Suppose thatX is a compact Riemann surface, prove thatH1(X,OX(−p)) ∼=H1(X,OX) for any p ∈ X.

5. Using the last exercise show that H1(X,O(q− p)) = 0 for any p, q if X hasgenus 0. Then conclude that OX(q) is globally generated. Show that thecorresponding map X → P1 is an isomorphism.

6. The euler characteristic χ(OX(D)) = dimH0(X,OX(D))−dimH1(X,OX(D)).Show that χ(O(D + p)) = χ(O(D)) + 1.

7. We stated a version of the Riemann-Roch theorem above. The actualtheorem says that for any divisor D on a genus g compact Riemann surface,χ(OX(D)) = degD + 1− g. Prove this by induction.

6.6. Function Fields and Automorphisms

Fix a smooth projective curve X over C. By the results of the last section, allcompact Riemann surfaces are of this form. The function field C(X) is the fieldof rational functions on X, or equivalently, by the exercises of §15.5, the field ofmeromorphic functions on it. Since C(X) = C(U) for any affine open set, we cansee that this field is finitely generated extension of C with transcendence degreeone. Let us refer to such fields as function fields. We claim conversely, that anysuch field F arises this way. We can write F as an algebraic extension of C(x).By the primitive element theorem [Ln], it is necessarily obtained by adjoining oneelement say y satisfying a polynomial f ∈ C(x)[t]. After clearing denominators, wehave f(x, y) = 0 with f ∈ C[x, t]. Thus F is the function field of the curve V (f).After replacing this by the normalization of its projective closure, we see that F isthe function field of a smooth projective curve. A more refined argument yields thefollowing equivalence:

Theorem 6.6.1. There is an anti-equivalence between the category of smoothprojective curves and surjective morphisms, and the category of function fields andinclusions.

Two categories are (anti-)equivalent if there is a (contravariant) functor induc-ing an isomorphism of Hom’s and if every object in the target is isomorphic to anobject in the image.

Proof. The details can be found in [Har, I, 6.12], but it is worth under-standing some key steps. A point of a smooth projective curve X gives a discretevaluation v = ordp : C(X)∗ → Z, which means that it satisfies

1. v(fg) = v(f) + v(g).2. v(f + g) ≥ min(v(f), v(g)).

In fact, any surjective discrete valuation is of this form for a unique p. Thus we canrecover the points from the function field. To see that an inclusion C(Y ) ⊆ C(X)

6.6. FUNCTION FIELDS AND AUTOMORPHISMS 95

of fields corresponds to a map f : X → Y , we can proceed as follows. A surjectivediscrete valuation v on C(X) yields discrete valuation on the subfield by restriction.This can be made surjective by rescaling it to v′ = 1

ev|C(Y )∗ for a unique integere > 0. Then v 7→ v′ determines a set theoretic map X → Y , which can be checkedto be a morphism.

Corollary 6.6.2 (Luroth’s theorem). Any subfield of C(x) properly containingC is abstractly isomorphic to C(x).

Proof. Let L ⊆ C(x) be such a field. It is necessarily a function field. There-fore L is the function field of some curve Y , and the inclusion corresponds to asurjective morphism f : P1 → Y . The following lemma implies that Y has genus 0,and is therefore isomorphic to P1 by the previous exercises.

Lemma 6.6.3. If f : X → Y is a surjective holomorphic map between compactRiemann surfaces, the genus of Y is less or equal to the genus of X.

Proof. If α is a nonzero holomorphic 1-form on Y , f∗α is easily seen to benonzero. Therefore, there is an injection H0(X,Ω1

X) → H0(Y,Ω1Y ).

The set of holomorphic automorphisms of a compact Riemann surface X formsa group Aut(X), which can be identified with the group of automorphisms of C(X)fixing C. For X = P1, this group is infinite since it contains (and is in fact equalto) PGL2(C). Likewise, for an elliptic curve X = C/L, Aut(X) is infinite since itcontains translations. For curves of larger genus, we have

Theorem 6.6.4 (Hurwitz). If X is a compact Riemann surface of genus g ≥ 2,Aut(X) is a finite group of cardinality at most 84(g − 1).

We want to give a proof of the finiteness part, since it gives a nice applicationof the ideas developed so far. Given an automorphism f : X → X, let Γf ⊂ X ×Xbe its graph. The Lefschetz trace formula 5.6.1 computes the intersection numberof Γ and the diagonal as an alternating sum of traces of the induced maps f∗ :Hi(X,R) → Hi(X,R). For i = 0, 2, these maps are easily seen to be the identity,thus

Γf ·∆ = 2− trace[f∗ : H1(X,R)→ H1(X,R)]

Then we claim that

Lemma 6.6.5. If f is different from the identity, then Γf ·∆ ≥ 0

Proof. The point is that here is that Γf ·∆ can be computed as a sum of inter-section numbers which are necessarily nonnegative. For transversal intersections,this was discussed in Section 5.5 and its exercises. The general case will treated inSection 12.2.

Corollary 6.6.6. If X has genus g ≥ 2, then f is the identity if and only iff∗ : H1(X,R)→ H1(X,R) is the identity.

Proof. If f∗ is the identity, then Γf ·∆ < 0. The other direction is clear.

Proof of theorem. It follows that there is an injective homomorphismAut(X)→Aut(H1(X,R)), and it suffices to prove that the image is finite. We do this by prov-ing that the group is both compact and discrete. For discreteness we note that for


any automorphism, f∗ preserves the lattice H1(X,Z). After choosing a basis ly-ing in H1(X,Z), this amounts to asserting that Aut(X) lies in GL2g(Z), which iscertainly a discrete subgroup of GL2g(R).

The automorphisms f∗ preserves the Hermitean pairing

〈α, β〉 =∫X

α ∧ β

on H1(X,R). This pairing will be shown positive definite in the exercises of Section9.1. Therefore Aut(X) lies in the corresponding unitary group which is compact.

Exercises

1. Prove that Aut(P1) = AutC(C(x)) = PGL2(C) acting by Mobius transfor-mations x 7→ ax+b

cx+d .2. Classify the discrete valuations on C(x) and show that they are all of the

form e · ordp for a unique e ∈ 1, 2, 3, . . . and p ∈ P1. (Hint: first showthat any discrete valuation on C is zero.)

3. If X is as in Hurwitz’s theorem with G = Aut(X), then Y = X/G canbe made into a compact Riemann surface of genus say h. At the levelof function fields C(Y ) = C(X)G. A special case of Riemann-Hurwitz’sformula gives

2g − 2 = |G|(2h− 2) +∑x

(|stab(x)| − 1)

where stab(x) is the stabilizer of x. Assuming this, show that (2g−2)/|G| ≥1/42, and thus complete the proof of the theorem.

4. Given a finite group G, construct a compact Riemann surface X with G ⊆Aut(X). (It’s enough to do this for the symmetric group.)

6.7. Modular forms and curves

Let Γ ⊂ SL2(R) be a subgroup acting properly discontinuously on H such thatH/Γ is compact. Let k a positive integer. An automorphic form of weight 2k is aholomorphic function f : H → C on the upper half plane satisfying

(6.7.1) f(z) = (cz + d)−2kf(az + b

cz + d)

for each (a bc d

)∈ Γ

Choose a weight 2k automorphic form f . Then f(z)(dz)⊗k is invariant under thegroup precisely when f is automorphic of weight 2k. Let us suppose that the groupΓ = Γ/±I acts freely, then the quotient X = H/Γ is a Riemann surface, andan automorphic form of weight 2k descends to a section of the sheaf ω⊗kX . We canapply Theorem 6.5.5 to the calculate the dimensions of these spaces.

6.7. MODULAR FORMS AND CURVES 97

Proposition 6.7.1. Suppose that Γ acts freely on H and that the quotientX = H/Γ is compact of genus g. Then the dimension of the space of automorphicforms of weight 2k is

g if k = 1(g − 1)(2k − 1) if k > 1

Proof. When k = 1, this is clear. When k > 1, we have

h0(ω⊗k) = h0(ω((k − 1)K)) = (k − 1)(deg K) + g − 1 = (2k − 1)(g − 1)

The above conditions are a bit too stringent, since they exclude some of themost interesting examples such as the modular group SL2(Z) and its subgroups.One such subgroup is the nth principal congruence group

Γ(n) =(

a bc d

)∈ SL2(Z) | a− 1 ≡ d− 1 ≡ b ≡ c ≡ 0 mod n

The quotient H/SL2(Z) can be identified with C as we saw. The natural

compactification P1C can be constructed as a quotient as follows. H corresponds

to the upper hemisphere of P1C = S2, and R ∪ ∞ corresponds to the equator.

We take H and add rational points on the boundary H∗ = H ∪ Q ∪ ∞. Theseare called cusps. Then SL2(Z) acts on H∗. The cusps form a single orbit, so wemay identify H∗/SL2(Z) ∼= P1 by sending this orbit to the point at infinity in P1.We can define a topology on H∗ by pulling back the open sets under the quotientmap π : H∗ → P1. This makes π continuous. It follows that H∗/Γ is a topologicalspace with a continuous projection to P1 for any subgroup Γ ⊆ SL2(Z). In fact,somewhat more is true.

Theorem 6.7.2. Given a finite index subgroup Γ ⊆ SL2(Z), H∗/Γ can bemade into a compact Riemann surface, called a modular curve, such that H∗/Γ→H∗/SL2(Z) ∼= P1 is holomorphic.

Proof. See [DS, §2.4].

Modular curves are of fundamental importance in number theory. A moreimmediate application is the construction of Riemann surfaces with large auto-morphism groups. When Γ is normal, then PSL2(Z)/Γ lies in Aut(H∗/Γ) andH∗/Γ → H∗/SL2(Z) is the quotient map. This is discussed further in the exer-cises.

A modular form of weight 2k for Γ ⊆ SL2(Z) is a holomorphic function f : H →C satisfying (6.7.1) and certain growth conditions which forces it to correspondto a holomorphic (or possibly meromorphic) object on H∗/Γ. For example, theEisenstein series

G2k(z) =∑

(m,n)∈Z2−(0,0)

1(m+ nz)2k

are modular of weight 2k for SL2(Z) if k > 1. Their significance for elliptic curvesis that g2 = 60G4(τ), g3 = 140G6(τ) give coefficients of the Weierstrass equation(6.2.1) for the lattice Z + Zτ .

As we saw, the points of the quotient H/SL2(Z) correspond to isomorphismclasses of elliptic curves. More generally the points of the quotient H/Γ(n) corre-spond to elliptic curves with some extra structure. Let us spell this out for n = 2.


An elliptic curve is an Abelian group, and a level two structure is a minimal set ofgenerators for its subgroup of 2-torsion elements.

Proposition 6.7.3. There is bijection between(a) The points of H/Γ(2).(b) The set of elliptic curves with level two structure.(c) P1 − 0, 1,∞.Proof. Given τ ∈ H, we get an elliptic curve Eτ = C/(Z + Zτ) with the level

two structure (1/2, τ/2) mod Z + Zτ . If τ ′ lies in the orbit of τ under Γ(2), thenthere is an isomorphism Eτ ∼= Eτ ′ taking (1/2, τ/2) mod Z + Zτ to (1/2, τ ′/2)mod Z + Zτ ′. Furthermore any elliptic curve is isomorphic to an Eτ , and the iso-morphism can be chosen so that a given level two structure goes over to the standardone. Thus H/Γ(2) classifies elliptic curves with level two structure as claimed. Wecan describe the set of such curves in another way. Given an elliptic curve E definedby Legendre’s equation (6.2.2), with t ∈ P1−0, 1,∞, the ramification points (thepoints on E lying over 0, 1,∞, t ) are precisely the 2-torsion points. We can takethe ramification point at ∞ to be the origin, then any other pair of branch pointsdetermines a level two structure Conversely, given an elliptic curve E, with origino and a level two structure (p, q), we have h0(OE(2o)) = 2. This means that thereis a meromorphic function f : E → P1

C with a double pole at o. It is not hardto see that f is ramified precisely at the two torsion points o, p, q, p + q (+ refersgroup law on E). By composing f with a (unique) automorphism of P1

C, we canput E in Legendre form such that f is projection to the x-axis, and f(o) = ∞,f(p) = t ∈ P1 − 0, 1,∞, f(q) = 0, f(p + q) = 1. Thus H/Γ(2) is isomorphic toP1 − 0, 1,∞.

We note that Γ(2)/±I acts freely on H, and the quotient is isomorphic toP1 − 0, 1,∞ as a Riemann surface. A nice consequence of all of this is:

Theorem 6.7.4 (Picard’s little theorem). An entire function omitting two ormore points must be constant.

Proof. The universal cover of P1 − 0, 1,∞ is H which is isomorphic to theunit disk D. Let f be an entire function omitting two points, which we can assumeare 0 and 1. Then f lifts to holomorphic map C → D which is bounded andtherefore constant by Liouville’s theorem.

ExercisesLet p be an odd prime in the following exercises.

1. Show that the stabilizers of the action of PSL2(Z) on points of H are trivialunless they lie in orbit of i or ρ = e2πi/3, in which case they are cyclic oforder 2 and 3 respectively. (Hint: refer back to the fundamental domain infigure 3.)

2. If Γ ⊂ PSL2(Z) is a finite index normal subgroup, show that π : H∗/Γ→H∗/SL2(Z)

j= P1 is a Galois cover with Galois group G = PSL2(Z)/Γ.

Show that the branch points of π are the images j(i), j(ρ),∞ in P1.3. Show that |PSL2(Z/pZ)| = p3−p

2 = dp. Check that the order of the stabiliz-ers of the PSL2(Z/pZ)-action on H∗/Γ(p) at points lying over j(i), j(ρ),∞are 2, 3 and p respectively.

6.7. MODULAR FORMS AND CURVES 99

4. Use the Riemann-Hurwitz formula (as stated in the exercises of §6.6), tocalculate that the genus of H∗/Γ(p) is

g = 1 +dp(p− 6)

12pShow that Hurwitz’s bound is attained with this example when p = 7.

CHAPTER 7

Simplicial Methods

In this chapter, we introduce simplicial methods which can be used to giveconcrete realizations for many of the cohomology theories encountered so far. Westart with simplicial (co)homology, which is historically the first such theory. It ishighly computable, at least in principle. Simplicial cohomology provides a modelfor Cech cohomology for arbitrary sheaves which is dealt with toward the end. Formany problems, Cech cohomology is the most convenient theory to work with. Inparticular, line bundles will be described very naturally using the Cech viewpoint.

7.1. Simplicial and Singular Cohomology

A systematic development of the ideas in this section can be found in [Sp].The standard n-simplex is

∆n = (t1, . . . tn+1) ∈ Rn+1 |∑

ti = 1, ti ≥ 0

The ith face ∆ni is the intersection of ∆n with the hyperplane ti = 0 (see (1). Each

face is homeomorphic to an n − 1 simplex, and this can be realized by an explicitaffine map δi : ∆n−1 → ∆n

i . In general, we refer to the intersection of ∆ with thelinear space ti1+1 = . . . tik+1 = 0 as a face. The faces are labeled by nonemptysubsets of 0, . . . n.

∆

∆3

3

3

Figure 1. 2 simplex

Some fairly complicated topological spaces, called polyhedra or triangulablespaces, can be built up by gluing simplices. It is known, although by no means

101

102 7. SIMPLICIAL METHODS

obvious, that manifolds and algebraic varieties (with classical topology) can betriangulated. The combinatorics of the gluing is governed by a simplicial complex.

Definition 7.1.1. A simplicial complex (V,Σ) consists of a set V , called theset of vertices, and a collection of finite nonempty subsets Σ of V containing all thesingletons and closed under taking nonempty subsets.

We can construct a topological space |(V,Σ)| out of a simplicial complex roughlyas follows. To each maximal element S ∈ Σ, choose an n-simplex ∆(S), where n+1is the cardinality of S. Glue ∆(S) to ∆(S′) along the face labeled by S∩S′ wheneverthis is nonempty. (When V is infinite, this gluing process requires some care, see[Sp, chap. 3].)

Let K = (V,Σ) be a simplicial complex, and assume that V is linearly ordered.We will refer to element of Σ as a n-simplex if it has cardinality n + 1. We definean n-chain on a simplicial complex to be a finite formal integer linear combination∑i ni∆i where the ∆i are n-simplices. In other words, the set of n-chains Cn(K) is

the free Abelian group generated by the set of n-simplices of K. Given an Abeliangroup, let Cn(K,A) = Cn(K) ⊗Z A. Dually, the set of n-cochains with valuesin A is Cn(K,A) = Hom(Cn(K), A). In other words, an n-cochain is functionwhich assigns an element of A to every n-simplex. One can think of a n-cochainsome sort of combinatorial analogue of an integral of n-form. As in integrationtheory we need to worry about orientations, and this is where the ordering comesin. An alternative, which is probably more standard, is to use oriented simplices;the complexes one gets this way are bigger, but the resulting cohomology theory isthe same.

We define the ith face of a simplex

δi(v0, . . . vn) = v0, . . . vi . . . vnv0 < v1 . . . < vn. (The notation x means omit x.) Given an n-chain C =

∑j aj∆j ,

we define a (n− 1)-chain ∂(C). called the boundary of C by

δ(C) =∑∑

(−1)i ajδi(∆j)

This operation can be extended by scalars to C•(K,A) and it induces a dual oper-ation ∂ : Cn(K,A)→ Cn+1(K,A) by ∂(F )(C) = F (δC). The key relation is

Lemma 7.1.2. δjδi = δiδj−1 for i < j.

An immediate corollary is:

Corollary 7.1.3. δ2 = 0 and ∂2 = 0.

Thus we have a complex. The simplicial homology of K is defined by

Hn(K,A) =ker[δ : Cn(K,A)→ Cn−1(K,A)]im[δ : Cn+1(K,A)→ Cn(K,A)]

Elements of the numerator are called cycles, and elements of the denominator arecalled boundaries. The simplicial cohomology are defined likewise by

Hn(K,A) = Hn(C•(K,A)) =ker[∂ : Cn(K,A)→ Cn+1(K,A)]im[∂ : Cn−1(K,A)→ Cn(K,A)]

.

Note that when V is finite, these groups are automatically finitely generated andcomputable. The choices of A of interest to us are Z,R, and C. The relationshipsare given by:

7.1. SIMPLICIAL AND SINGULAR COHOMOLOGY 103

Theorem 7.1.4 (Universal coefficient theorem). If A is torsion free, then thereare isomorphisms

Hi(K,A) ∼= Hi(K,Z)⊗A

Hi(K,A) ∼= Hi(K,Z)⊗A ∼= Hom(Hi(K,Z), A)

Proof. [Sp].

One advantage of cohomology over homology is that it has a multiplicativestructure. When A is a commutative ring R, there is a product on cohomology anal-ogous to the product in De Rham induced by wedging forms. Given two cochainsα ∈ Cn(K,R), β ∈ Cm(K,R), their cup product α ∪ β ∈ Cn+m(K,R) is given by

(7.1.1) α ∪ β(v0, . . . vn+m) = α(v0, . . . vn)β(vn, . . . vn+m)

where v0 < v1 < . . .. Then:

Lemma 7.1.5. ∂(α ∪ β) = ∂(α) ∪ β + (−1)nα ∪ ∂(β)

Corollary 7.1.6. ∪ induces an operation on cohomology that makes H∗(K,R)into a graded ring.

Singular (co)homology was introduced partly in order to give conceptual proofof the fact that Hi(K) and Hi(K) depends only on |K|, i.e. that these are indepen-dent of the triangulation. Here we concentrate on singular cohomology. A singularn-simplex on a topological space X is simply a continuous map from f : ∆n → X.When X is a manifold, we can require the maps to be C∞. We define a singularn-cochain on a X be a map which assigns an element of A to any n-simplex on X.Let Sn(X,A) (Sn∞(X,A)) denote the group of (C∞) n-cochains with values in A.When F is an n-cochain, its coboundary is the (n+ 1)-cochain

∂(F )(f) =∑

(−1)iF (f δi).

The following has more or less the same proof as corollary 7.1.3.

Lemma 7.1.7. ∂2 = 0.

The singular cohomology groups of X are

Hising(X,A) = Hi(S∗(X,A)).

Singular cohomology is clearly contravariant in X. A basic property of this co-homology theory is its homotopy invariance. We state this in the form that wewill need. A subspace Y ⊂ X is called a deformation retraction, if there exists aa continuous map F : [0, 1] × X → X such that F (0, x) = x, F (1, X) = Y andF (1, y) = y for y ∈ Y . X is called contractible if it deformation retracts to a point.

Proposition 7.1.8. If Y ⊂ X is a deformation retraction, then

Hising(X,A)→ Hi

sing(Y,A)

is an isomorphism for any A.

In particular, the higher cohomology vanishes on a contractible space. This isan analogue of Poincare’s lemma. We call a space locally contractible if every pointhas a contractible neighbourhood . Manifolds and varieties with classical topologyare examples of such spaces.


Theorem 7.1.9. If X is paracompact Hausdorff space (e. g. a metric space)which is locally contractible, then Hi(X,AX) ∼= Hi

sing(X,A) for any Abelian groupA.

A complete proof can be found in [Sp, chap. 6] (note that Spanier uses Cechapproach discussed in the next chapter). In the case of manifolds, a proof whichis more natural from our point of view can be found in [Wa]. The key step is toconsider the sheaves Sn associated to the presheaves U 7→ Sn(U,A). These sheavesare soft since they are modules over the sheaf of real valued continuous functions.The local contractability guarantees that

0→ AX → S0 → S1 → . . .

is a fine resolution. Thus one gets

Hi(X,AX) ∼= Hi(S∗(X))

It remains to check that the natural map

S∗(X,A)→ S∗(X)

induces an isomorphism on cohomology. We the reader refer to [Wa, pp 196-197]for this.

As a corollary of this (and the universal coefficient theorem), we obtain theform of De Rham’s theorem that most people think of.

Corollary 7.1.10 (De Rham’s theorem, version 2). If X is a manifold,

HidR(X,R) ∼= Hi

sing(X,R) ∼= Hising(X,Z)⊗ R

The theorem and its corollary also holds when C∞ cochains are used. In thiscase, the map can be defined directly on the level of complexes by

α 7→ (f 7→∫

∆

f∗α)

Singular cohomology carries a cup product given by formula (7.1.1). A strongerform of De Rham’s theorem shows that the above map is a ring isomorphism [Wa].Fundamental classes of oriented submanifolds can be constructed in H∗(X,Z). Thiscan be used to show that the intersection numbers Y · Z are always integers. Thiscan also be deduced from proposition 5.5.3 and transervsality theory.

Exercises

1. Prove Lemma 7.1.2 and it corollary.2. Calculate the simplicial cohomology with Z coefficients for the “tetrahe-

dron” which is the powerset of V = 1, 2, 3, 4 with ∅ and V removed.3. Let Sn be the n-sphere realized as the unit sphere in Rn+1. Let U0 = Sn−(0, . . . 0, 1) and U1 = Sn−(0, . . . 0,−1). Prove that Ui are contractible,and that U0∩U1 deformation retracts on to the “equatorial” (n−1)-sphere.

4. Prove that

Hi(Sn,Z) =

Z if i = 0, n0 otherwise

using Mayer-Vietoris.

7.2. H∗(Pn,Z) 105

7.2. H∗(Pn,Z)

We can now carry out a basic computation. Let Pn = PnC with its classicaltopology.

Theorem 7.2.1.

Hi(Pn,Z) =

Z if 0 ≤ i ≤ 2n is even0 otherwise

Before giving the proof, we will need to develop a few more tools. Let X be aspace satisfying the assumptions of Theorem 7.1.9, and Y ⊂ X a closed subspacesatisfying the same assumptions. We will insert the restriction map

Hi(X,Z)→ Hi(Y,Z)

into a long exact sequence. This can be done in a number of ways, by definingcohomology of the pair (X,Y ), by using sheaf theory, or by using a mapping cone.We will choose the last option. Let C be obtained by first gluing the base of thecylinder 1 × Y ⊂ [0, 1]× Y to X along Y , and then collapsing the top to a pointP (figure (2)).

X

Y

C

P

Figure 2. Mapping cone

Let U1 = C − P , and U2 ⊂ C the open cone [0, 1) × Y/0 × Y (the notationA/B means collapse B to a point). One sees that U1 deformation retracts to X,U2 is contractible, and U1 ∩ U2 deformation retracts to Y . The Mayer-Vietorissequence, together with proposition 7.1.8, yields a long exact sequence

. . . Hi(C,Z)→ Hi(X,Z)→ Hi(Y,Z)→ Hi+1(C,Z) . . .

when i > 0. To make this really useful, note that the map C → C/U2 whichcollapses the closed cone to a point is a homotopy equivalence. Therefore it inducesan isomorphism on cohomology. Since we can identify C/U2 with X/Y , we obtaina sequence

(7.2.1) . . . Hi(X/Y,Z)→ Hi(X,Z)→ Hi(Y,Z)→ . . .


We apply this when X = Pn and Y = Pn−1 embedded as a hyperplane. Thecomplement X − Y = Cn. Collapsing Y to a point amounts to adding a point atinfinity to Cn, thus X/Y = S2n. Since projective spaces are connected

H0(Pn,Z) ∼= H0(Pn−1,Z) ∼= Z.For i > 0, (7.2.1) and the previous exercise yields isomorphisms

(7.2.2) Hi(Pn,Z) ∼= Hi(Pn−1,Z), when i < 2n

H2n(Pn,Z) ∼= ZTheorem 7.2.1 now follows by induction.

Exercises

1. Let L ⊂ Pn be a linear subspace of codimension i. Prove that its funda-mental class [L] generated H2i(Pn,Z).

2. Let X ⊂ Pn be a smooth projective variety. Then [X] = d[L] for some d,where L is a linear subspace of the same dimension. d is called the degreeof X. Bertini’s theorem, which you can assume, implies that there exists alinear space L′ of complementary dimension transverse to X. Check thatX · L′ = #(X ∩ L′) = d.

3. An algebraic variety X admits an affine cell decomposition if there is ansequence of Zariski closed sets X = Xn ⊃ Xn−1 . . . such that each differenceXi −Xi−1 is an affine space. Generalize the above procedure to computeH∗(X,Z) assuming that X possesses such a decomposition.

4. Let G = G(2, 4) be the Grassmanian of lines in P3. We define a sequenceof subsets by imposing so called Schubert conditions. Fix a plane P ⊂ P3,a line L ⊂ P and a point Q ∈ L. Let G3 be the set of lines meeting L,G2 the set of lines containing Q or contained in P , G1 ⊂ G the of linescontaining Q and contained in P , and let G0 = L. Show that this givesan affine cell decomposition. Use this to calculate the cohomology of G.

7.3. Cech cohomology

We return to sheaf theory proper. We will introduce the Cech approach tocohomology which has the advantage of being quite explicit and computable (andthe disadvantage of not always giving the “right” answer). Roughly speaking,Cech bears the same relation to sheaf cohomology, as simplicial does to singularcohomology.

One starts with an open cover Ui | i ∈ I of a space X indexed by a linearlyordered set I. If J ⊆ I, let UJ be the intersection of Uj with j ∈ J . Let F be asheaf on X. The group of Cech n-cochains is

Cn = Cn(Ui,F) =∏

i0<...<in

F(Ui0...in)

The coboundary map ∂ : Cn → Cn+1is defined by

∂(f)i0...in+1 =∑k

(−1)kfi0...ik...in+1|Ui0...in+1

.

By an argument similar to the proof of corollary 7.1.3, we have:

Lemma 7.3.1. ∂2 = 0

7.3. CECH COHOMOLOGY 107

Definition 7.3.2. The nth Cech cohomology group is

Hn(Ui,F) = Hn(C•(Ui,F)) =ker(∂ : Cn → Cn+1)im(∂ : Cn−1 → Cn)

To get a feeling for this, let us write out the first couple of groups explicitly:

H0(Ui,F) = (fi) ∈∏F(Ui) | fi = fj on Uij

= F(X)

(7.3.1) H1(Ui,F) =(fij) ∈

∏F(Uij) | fik = fij + fjk on Uijk

(fij) | ∃(φi), fij = φi − φjThere is a strong similarity with simplicial cohomology. This can be made

precise by introducing a simplicial complex called the nerve of the cover. For theset of vertices, we take the index set I. The set of simplices is given by

Σ = i0, . . . in |Ui0,...in 6= ∅

If we assume that each Ui0,...in is connected, then we see that the Cech complexCn(Ui, AX) coincides with the simplicial complex of the nerve with coefficientsin A.

Even though, we are primarily interested in sheaves of Abelian groups. It willbe convenient to extend (7.3.1) to a sheaf of arbitrary groups G.

H1(Ui,G) = (gij) ∈∏i<j

G(Uij) | gik = gijgjk on Uijk /∼

where (gij) ∼ (gij) if there exists (γi) ∈∏G(Ui) such that gij = γigijγ

−1j . Note

that in general this is not a group, but just a set with distinguished element gij = 1.The (gij) are called 1-cocycles with values in G. It will be useful to drop therequirement that i < j by setting gji = g−1

ij and gii = 1.As an example of a sheaf of nonabelian groups, take U 7→ GLn(R(U)), where

(X,R) is a ringed space (i.e. space with a sheaf of commutative rings).

Theorem 7.3.3. Let (X,R) be a manifold or a variety over k, and Ui andopen cover of X. There is a bijection between the following sets:

1. The set of isomorphism classes of rank n vector bundles over (X,R) triv-iallizable over Ui.

2. The set of isomorphism classes of locally free R-modules M such thatM |Ui

∼= R|nUi.

3. H1(Ui, Gln(R)).

Proof. We merely describe the correspondences.1→ 2: Given a vector bundle V on X, let V its sheaf of sections.2→ 3: Given a locally free sheaf V. Choose isomorphisms Fi : RnUi

→ V|Ui. Set

gij = Fi F−1j . This determines a well defined element of H1.

3→ 1: Define an equivalence relation ≡ on the disjoint union W =∐Ui × kn as

follows. Given (xi, vi) ∈ Ui × kn and (xj , vj) ∈ Uj × kn, (xi, vi) ≡ (xj , vj)if and only if xi = xj and vi = gij(x)vj . Let V = (W/≡) with quotienttopology. Given an open set (U ′/≡) = U ⊂ V . Define f : U → k to be


regular, C∞ or holomorphic (as the case may be) if its pullback to U ′ hasthis property.

Implicit above, is a construction which associates to a 1-cocycle γ = (gij), thelocally free sheaf

Mγ(U) = (vi) ∈∏R(U ∩ Ui)n | vi = gijvj.

Consider the case of projective space P = Pnk . Suppose x0, . . . xn are homogeneouscoordinates. Let Ui be the complement of the hyperplane xi = 0. Then Ui isisomorphic to Ank by

[x0, . . . xn]→ (x0

xi, . . .

xixi. . .)

Define gij = xj/xi ∈ O(Uij)∗. This is a 1-cocycle, and Mgij∼= OP(1). Likewise

(xj/xi)d is the 1-cocyle for O(d).We get rid of the dependence on covers by taking direct limits. If Vj is

refinement of Ui, there is a natural restriction map

Hi(Ui,F)→ Hi(Vj,F)

Definition 7.3.4. The nth Cech cohomology group

Hi(X,F) = lim→Hi(Uj,F)

Corollary 7.3.5. There is a bijection between the following sets:

1. The set of isomorphism classes of rank n vector bundles over (X,R).2. The set of isomorphism classes of locally free R-modules M of rank n.3. H1(X,Gln(R)).

A line bundle is a rank one vector bundle. We won’t distinguish between linesbundles and rank one locally free sheaves. The set of isomorphism classes of linebundles carries the structure of a group namely H1(X,R∗). This group is calledthe Picard group, and is denoted by Pic(X).

Exercises

1. Check that the Cech coboundary satisfies ∂2 = 0.2. Fill in the details of the proof of theorem 7.3.3.3. Calculate the cocycle corresponding to the tautological line bundle

L = (x, `) ∈ V × P | x ∈ ` → Pn

and check that coincides with the formula for OP(−1) given above.4. Let U = Pn+1−[0, . . . , 0, 1], and let U → Pn be the projection [x0, . . . xn+1] 7→

[x0, . . . xn]. Prove that U → Pn is a line bundle. Calculate the cocycle, andshow that it corresponds to OP(1).

5. Show that multiplication in Pic(X) can be interpreted as tensor productof line bundles.

7.4. CECH VERSUS SHEAF COHOMOLOGY 109

7.4. Cech versus sheaf cohomology

We define a sheafified version of the Cech complex. Given a sheaf F on a spaceX, let

Cn(Ui,F) =∏

i0<...<in

ι∗F|Ui0...in

where ι denotes the inclusion Ui0...in ⊂ X. We construct a differential ∂ using thesame formula as in the usual Cech complex defined earlier. Taking global sectionsyields the usual Cech complex.

Lemma 7.4.1. These sheaves fit into an exact sequence

0→ F → C0(Ui,F)→ C1(Ui,F) . . .

Proof. [Har, III, Lemma 4.2].

Lemma 7.4.2. Suppose that F is flasque, then Hn(Ui,F) = 0 for all n > 0.

Proof. If F is flasque, then the sheaves Cn(Ui,F) are also seen to be flasque.Then this gives an acyclic resolution of F . Therefore

Hn(Ui,F) = Hn(C•(Ui,F)) = Hn(X,F) = 0.

Lemma 7.4.3. Suppose that H1(UJ ,A) = 0 for all nonempty finite sets J . Thengiven an exact sequence

0→ A→ B → C → 0of sheaves, there is a long exact sequence:

0→ H0(Ui,A)→ H0(Ui,B)→ H0(Ui, C)→ H1(Ui,A) . . .

Proof. The hypothesis guarantees that there is are short exact sequences

0→ A(UJ)→ B(UJ)→ C(UJ)→ 0

Thus we have an exact sequence of complexes:

0→ C•(Ui,A)→ C•(Ui,B)→ C•(Ui, C)→ 0.

The long exact sequence now follows from a standard result in homological algebra(c.f. [Wl, theorem 13.1]).

Definition 7.4.4. An open cover Ui is called a Leray cover for a sheaf F ifHn(UJ ,F) = 0 for all nonempty finite sets J and all n > 0.

Theorem 7.4.5. If Ui is a Leray cover for the sheaf F , then

Hn(Ui,F) ∼= Hn(X,F)

for all n.

Proof. This is clearly true for n = 0. With the notation of Section 4.2, wehave an exact sequence

0→ F → G(F)→ C1(F)→ 0

withH1(X,F) = coker[Γ(X,G(F))→ Γ(X,C1(F))]

andHn+1(F) = Hn(C1(F)).


Lemmas 7.4.3 and 7.4.2 imply that

H1(F) ∼= coker[Γ(X,G(F))→ Γ(X,C1(F))]

and

Hn+1(F) = Hn(C1(F))

for n > 0. This already proves the theorem for n = 1. The remaining cases forn > 1 can be handled by induction, because the cover is also Leray with respect toC1(F):

Hi(UJ , C1(F)) = Hi+1(UJ ,F) = 0

Corollary 7.4.6. If every cover admits a Leray refinement, then Hn(X,F) ∼=Hn(X,F).

We state a few more general results for the record.

Proposition 7.4.7. For any sheaf F

H1(X,F) ∼= H1(X,F)

Proof. See [G, Cor. 5.9.1]

Theorem 7.4.8. If X is a paracompact Hausdorff space (e.g. a metric space),then for any sheaf and all i,

Hi(X,F) ∼= Hi(X,F)

for all i.

Proof. See [G, Cor. 5.10.1]

Exercises

1. If Ui is an open cover of space X such that every nonempty intersectionUI is contractible, show that it is Leray with respect to a constant sheafAX . Thus Hi(X,AX) is the simplicial cohomology of the nerve associatedto such a cover.

2. A finite dimensional cell complex, which refines the notion of a simplicialcomplex, is a space constructed as follows. Start with a set of points X0,then glue a set of intervels (1-cells) by attatching their end points to X0

to obtain X1. Glue a set of 2-disks (2-cells) by attaching their boundariesto X1 and so on. X = Xn for some n. Show that X admits an open coveras in the previous exercise.

3. When Uj is a finite open cover, use Mayer-Viteoris and induction toconstruct a natural map Hi(Uj,F) → Hi(X,F), which yields an iso-morphism when the cover is Leray.

7.5. FIRST CHERN CLASS 111

7.5. First Chern class

Let (X,OX) be a complex manifold or algebraic variety over C. Then we haveisomorphisms

Pic(X) ∼= H1(X,O∗X) ∼= H1(X,O∗X)

Let e(x) = exp(2πix) be the normalized exponential.

Lemma 7.5.1. When X is a complex manifold, there is an exact sequence

(7.5.1) 0→ ZX → OXe−→ O∗X → 1

called the exponential sequence.

Proof. It suffices to check this when X is replaced by a ball. Since a ballis simply connected, given a holomorphic function f ∈ O∗(X), we can find a sin-gle valued branch of the logarithm log f . Then f = e(log f/2πi). This provessurjectivity. The remaining steps are straightforward.

Definition 7.5.2. Given a line bundle L, its first Chern class c1(L) ∈ H2(X,Z)is the image of L under the connecting map Pic(X)→ H2(X,Z)

This can be carried out for C∞ manifolds as well, provided one interprets OXas the sheaf of complex valued C∞ functions, and Pic(X) as group of C∞ complexline bundles. In this case, c1 is an isomorphism. It is clear that the construction isfunctorial:

Lemma 7.5.3. If f : X → Y is C∞ map between manifolds, c1(f∗L) = f∗c1(L).

We want to calculate the first Chern class explicitly for a compact Riemannsurface X. Note that in this case H2(X,Z) ∼= Z, so we can view c1 as a number.To make things even more explicit, we note that this isomorphism is given by theinclusion H2(X,Z) ⊂ H2(X,C) followed by integration. In order to get a de Rhamrepresentative, we consider the commutative diagram

0 // ZX //

OXe //

O∗X //

d log /2πi

1

0 // CX // E0X

d // E1X,cl

// 0

where E1X,cl is the sheaf of closed 1-forms. By definition, it fits into an exact sequence

0→ E1X,cl → E1

X → E2X → 0

From this we obtain

E1(X)→ E2(X)→ H1(X, EX,cl)→ 0

Given a 1 cocycle γjk representing an element on the right, we can find an explicitlift to β ∈ E2(X) as follows. Regarding γjk as a cocycle in E1

X , we have

γjk = αj |Ujk− αk|Ujk

, αj ∈ E1X(Uj)

because E1X is soft. Since γjk is closed, we have dαj = dαk on Ujk. Thus these

forms patch to give a global 2 form β.


From the previous diagram, we obtain a commutative square

H1(X,O∗X)c1 //

d log /2πi

H2(X,Z)

H1(X, EX,cl) // H2(X,C)

Combining all of this leads to the following recipe for computing c1. Apply d log2πi to

a 1-cocycle gjk ∈ O∗X(Ujk). Then write it as a boundary:

d log gjk2πi

= αj |Ujk− αk|Ujk

, αj ∈ E1X(Uj)

Then dαj patches to give a well defined global closed 2-form β which represents theimage of c1 in H2(X,C). Finally, we integrate β to get a number.

Lemma 7.5.4. On P1 with the above identification, we have c1(O(1)) = 1.

Proof. Set P = P1C. We can use the standard cover Ui = xi 6= 0. We

identify U1 with C with the coordinate z = x0/x1. The 1-cocycle (section 7.3) ofOX(1) is g01 = z−1. Carrying out the above procedure, yields

d log g01

2πi= − dz

2πiz= α0 − α1

and β = dαj .In order to evaluate integral

∫β divide the sphere into two hemispheres H1 =

|z| ≤ 1 and H0 = |z| ≥ 1. Let C be the curve |z| = 1 oriented so that theboundary of H1 is C. Then with the help of Stokes’ theorem, we get∫

Pβ =

12πi

(∫H0

dα0 +∫H1

dα1)

= − 12πi

(∫C

α0 −∫C

α1)

=1

2πi

∫C

dz

z= 1

Thus c1(O(1)) is the fundamental class of H2(P1,Z).

By the same kind of argument, we obtain:

Lemma 7.5.5. If D is a divisor on a compact Riemann surface X, c1(O(D)) =deg(D)

We are going to generalize this to higher dimensions. A complex submanifoldD ⊂ X of a complex manifold is called a smooth effective divisor if D is locallydefinable by a single equation. In other words, we have an open cover Ui of X,and functions fi ∈ O(Ui), such that D ∩ Ui is given by fi = 0. We define theOX -module OX(−D) to be the ideal sheaf of D, and OX(D) to be the dual. Byassumption, OX(−D) is locally a principle ideal, and hence a line bundle. The linebundle OX(D) is determined by the 1-cocycle fi/fj ∈ O(Ui ∩ Uj)∗.

Lemma 7.5.6. If H ⊂ Pn is a hyperplane, then OPn(H) ∼= OPn(1).

7.5. FIRST CHERN CLASS 113

Proof. Let H be given by the homogeneous linear form ` =∑k akxk = 0.

Then for the standard cover Ui = xi 6= 0, H is defined by

`i =∑

akxkxi

= 0.

Thus O(H) is determined by the 1-cocycle `i/`j = xj/xi, which is the cocycle forO(1).

Theorem 7.5.7. If D is a smooth effective divisor, then c1(OX(D)) = [D].

We start with the special case.

Lemma 7.5.8. c1(OPn(1)) = [H] where H ⊂ Pn is a hyperplane.

Proof. We have already checked this for P1 in Lemma 7.5.4. Embed P1 ⊂ Pnas a line. The restriction map induces an isomorphism on second cohomology withZ coefficients such that [H] maps to the fundamental class of P1 by (7.2.2). Sincec1 is compatible with restriction, Lemma 7.5.4 implies that c1(OPn(1)) = [H].

Proof of theorem 7.5.7. There are several ways to prove this. We are goingto outline the proof given in [Hrz] which is reduces it to the previous lemma. Weare going to work in the C∞ category. Most of the constructions used here cannotbe made holomorphic. Choose an open tubular neighbourhood T of D. This is acomplex C∞ line bundle such that D corresponds to the zero section. So there isa C∞ classifying map X → Pn such that the line bundle associated to OPn(−1)pulls back to T 2.6.4. By composing this with complex conjugation Pn → Pn, wecan arrange that the dual line bundle U associated to OPn(1) pulls back to T (seeexercises).

The Thom space Th(M) of a line bundle M → X is just the one point com-pactification of M . The Thom space Th(T ) can simply be obtained by collapsingthe complement of T in X to a point. From the exercises of Section 7.3, we can seethat Th(U) can be identified with Pn+1. Under this identification, the zero sectioncorresponds to Pn embedded as a hyperplane. Since Th(T ) will map to Th(L), weobtain a commutative diagram.

D //

##FFFFFFFFF X

Th(T )

Pn // Pn+1

Using this, we can see that the fundamental class of Pn in Pn+1 pulls back to [D].Furthermore the C∞ line bundle associated OX(D) is isomorphic to the pullbackof the C∞ line bundle associated to OPn+1(1). This is because the duals can beidentified with the ideal sheaf of D and Pn respectively. Thus the theorem followsfrom the previous lemma.

Exercises

1. Prove Lemma 7.5.5.


2. Given a vector bundle V of rank r, define det(V ) = ∧rV and c1(V ) =c1(detV ). Prove that det(V1⊕V2) ∼= det(V1)⊗det(V2). Use this to calculatec1(V1 ⊕ V2).

3. A Hermitean metric || ||2 on a holomorphic line bundle L→ X is complexvalued C∞ function on the total space L which restricts to an inner producton the fibres. Every line possesses a metric by using a partition of unity.Conclude that L can represented by a cocycle gij with values in the unitcircle. Use this to show that the pullback L of L under complex conjugationis isomorphic to the dual L∗.

4. Given Hermitean metric on L and a local trivialization φi : L|Ui∼= Ui ×

C, let hi = ||φ−1i (1)||2. This gives a collection of positive C∞ functions

satisfying hi = ||gij ||2hj . Show that the curvature ω = ∂∂hi is globallydefined, and that 1

2π√−1ω represents c1(L). (See (9.2.1) for a discussion of

the ∂, ∂ operators.)

Part 3

Hodge Theory

CHAPTER 8

The Hodge theorem for Riemannian manifolds

Thus far, our approach has been primarily algebraic or topological. We aregoing to need a basic analytic result, namely the Hodge theorem, which says thatevery de Rham cohomology class has a unique “smallest” element, called its har-monic representative. Standard accounts of basic Hodge theory can be found inthe books of Griffiths-Harris [GH], Warner [Wa] and Wells [W]. However, we willdepart slightly from these treatments by outlining the heat equation method ofMilgram and Rosenbloom [MR, BGV]. This is an elegant and comparatively ele-mentary approach to the Hodge theorem. As a warm up, we will do a combinatorialversion which requires nothing more than linear algebra.

8.1. Hodge theory on a simplicial complex

In order to motivate the general Hodge theorem, we first work out a simplecombinatorial analogue. Let K = (V,Σ) be a finite simplicial complex. Then thespaces of cochains C∗(K,R) are finite dimensional. For each simplex S, let

δS(S′) =

1 if S = S′

0 otherwise

These form a basis of C∗(K,R). Now choose inner products on these spaces. Aparticularly natural choice is determined by making this basis orthonormal. Let∂∗ : Ci(K,R) → Ci−1(K,R) be the adjoint to ∂. Then ∆ = ∂∂∗ + ∂∗∂ is thediscrete Laplacian.

Lemma 8.1.1. Let α be a cochain. The following are equivalent:(1) α ∈ (im ∂)⊥ ∩ ker ∂.(2) ∂α = ∂∗α = 0.(3) ∆α = 0.

Proof. We prove the equivalences of (1) and (2), and (2) and (3). Supposethat α ∈ (im ∂)⊥ ∩ ker ∂. Then of course ∂α = 0. Furthermore, since

||∂∗α||2 = 〈α, ∂∂∗α〉 = 0

it follows that ∂∗α = 0. Conversely, assuming (2), we have α ∈ ker ∂ and

〈α, ∂β〉 = 〈∂∗α, β〉 = 0.

If ∂α = ∂∗α = 0, then ∆α = 0. Finally suppose that ∆α = 0. Then

〈∆α, α〉 = ||∂α||2 + ||∂∗α||2 = 0

which implies (2).

The cochains satisfying the above conditions are called harmonic.

117

118 8. THE HODGE THEOREM FOR RIEMANNIAN MANIFOLDS

Lemma 8.1.2. Every simplicial cohomology class has a unique harmonic repre-sentative. The harmonic representative minimizes norm in its cohomology class.

Proof. To prove the first statement, we check that the map

h : (im ∂)⊥ ∩ ker ∂ → Hn(K,R)

sending a harmonic cochain to its cohomology class is an isomorphism. If α lies inthe kernel of h then it is a coboundary. Therefore α = ∂β which implies that

||α||2 = 〈α, ∂β〉 = 0

This shows that h is injective. Given a cochain α with ∂α = 0. We can decomposeit as α = α1 + α2 with α1 ∈ (im ∂)⊥ ∩ ker ∂ and α2 ∈ im ∂. Thus the cohomologyclass of α = h(α1).

Given a harmonic element α,

||α+ ∂β||2 = ||α||2 + ||∂β||2 > ||α||2

unless ∂β = 0.

Exercises

1. Prove that the space of cochains can be decomposed into an orthogonaldirect sum

Cn(K) = ker ∆⊕ im ∂ ⊕ im ∂∗

2. Prove that ∆ is a positive semidefinite symmetric operator.3. Use 2 to prove that limit of the “heat kernel”

H = limt→∞

e−t∆

exists and is the orthogonal projection to ker ∆.4. Let G be the endomorphism of Cn(K) which acts by 0 on ker ∆ and by 1

λon the λ-eigenspaces of ∆ with λ 6= 0. Check that I = H + ∆G.

8.2. Harmonic forms

Let X be an n dimensional compact oriented manifold. We want to prove ananalogue of lemma 8.1.2 for de Rham cohomology. In order to formulate this, weneed inner products. A Riemannian metric (, ), is a family of inner products onthe tangent spaces which vary in a C∞ fashion. This means that inner productsare determined by a tensor g ∈ Γ(X, E1

X ⊗ E1X). The existence of Riemannian

metrics can be proved using a standard partition of unity argument [Sk2, Wa].A Riemannian manifold is a C∞ manifold endowed with a Riemannian metric. Ametric determines inner products on exterior powers of the cotangent bundle whichwill also be denoted by (, ). In particular, the inner product on the top exteriorpower gives a tensor det(g) ∈ Γ(X, EnX ⊗ EnX). Since X is oriented it is possible tochoose a consistent square root of det(g) called the volume form dvol ∈ En(X). Inlocal coordinates

dvol =√

det(gij) dx1 ∧ . . . ∧ dxnwhere ∂

∂x1, . . . ∂

∂xnis positively oriented. The Hodge star operator is a C∞(X)-

linear operator ∗ : Ep(X)→ En−p(X), determined by

(8.2.1) α ∧ ∗β = (α, β)dvol

8.2. HARMONIC FORMS 119

After choosing a local orthonormal basis or frame ei for E1X in a neighbourhood of

a point, ∗ is easy to calculate, e.g. ∗e1 ∧ . . . ek = ek+1 ∧ . . . en. From this one cancheck that ∗∗ = ±1. The spaces Ep(X) carry inner products:

〈α, β〉 =∫X

(α, β)dvol =∫X

α ∧ ∗β

and hence norms.

Lemma 8.2.1. For all α, β,

〈dα, β〉 = ±〈α, ∗d ∗ β〉.In other words, the adjoint d∗ to d is ± ∗ d∗.

Proof. The proof follows by applying Stokes’ theorem to the identity

d(α ∧ ∗β) = dα ∧ ∗β ± α ∧ ∗ ∗ d ∗ β(This is nothing other integration by parts.)

The key result of this chapter is:

Theorem 8.2.2 (The Hodge theorem). Every de Rham cohomology class hasa unique representative which minimizes norm. This is called the harmonic repre-sentative.

To understand the meaning of the harmonicity condition, we can treat it asvariational problem and find the Euler-Lagrange equation. Let α be a harmonicp-form. Then for any (p− 1)-form β, we would have to have

d

dt||α+ tdβ||2|t=0 = 2〈α, dβ〉 = 2〈d∗α, β〉 = 0,

which forces d∗α = 0. Conversely if d∗α = 0 then

||α+ dβ||2 = ||α||2 + ||dβ||2

which implies that α has the minimum norm in its cohomology class. Thus har-monicity can be expressed as a pair of differential equations dα = 0 and d∗α = 0.It is sometimes more convenient to combine these into a single equation. For thiswe need

Definition 8.2.3. The Hodge Laplacian is ∆ = d∗d+ dd∗.

Lemma 8.2.4. The following are equivalent(1) α is harmonic.(2) dα = 0 and d∗α = 0.(3) ∆α = 0.

Proof. The equivalence of the first two conditions is contained in the previousdiscussion. The equivalence of the last two conditions follows from the identity〈∆α, α〉 = ||dα||2 + ||d∗α||2.

The hard work is contained in the following result, whose proof will be post-poned until the next section.

Theorem 8.2.5. There are linear operators H (harmonic projection) and G(Green’s operator) taking C∞ forms to C∞ forms, which are characterized by thefollowing properties


(1) H(α) is harmonic,(2) G(α) is orthogonal to the space of harmonic forms,(3) α = H(α) + ∆G(α),for any C∞ form α.

Corollary 8.2.6. There is an orthogonal direct sum

E i(X) = (harmonic forms)⊕ dE i−1(X)⊕ d∗E i+1(X)

Proof. Exercise.

Proof of theorem 8.2.2. We prove the existence part of the theorem. Theuniqueness is straightforward and left as an exercise. Let α be a closed form. Itcan be written as α = β + dd∗γ + d∗dγ with β = H(α) and γ = G(α). Since β isharmonic,

〈d∗dγ, β〉 = 〈dγ, dβ〉 = 0Furthermore,

〈d∗dγ, dd∗γ〉 = 〈d∗γ, d2d∗γ〉 = 0Thus

||d∗dγ||2 = 〈d∗dγ, α〉 = 〈dγ, dα〉 = 0Therefore α is cohomologous to the harmonic form β.

Theorem 8.2.5 remains, and this will be dealt with later. For now we considerthe easy, but instructive, example of the torus

Example 8.2.7. X = Rn/Zn, with the Euclidean metric. A differential formα can be expanded in a Fourier series

(8.2.2) α =∑λ∈Zn

∑i1<...<ip

aλ,i1...ipe2πiλ·xdxi1 ∧ . . . ∧ dxip

By direct calculation, one finds the Laplacian

∆ = −∑ ∂2

∂x2i

,

the harmonic projection

H(α) =∑

a0,i1...ipdxi1 ∧ . . . ∧ dxipand Green’s operator

G(α) =∑

λ∈Zn−0

∑ aλ,i1...ip4π2|λ|2

e2πiλ·xdxi1 ∧ . . . ∧ dxip .

Since the image of H are forms with constant coefficients, this proves the assertionin example 5.3.13.

As an application of this theorem, we have get a new proof of a Poincare dualityin strengthened form.

Corollary 8.2.8. (Poincare duality, version 2). The pairing

Hi(X,R)×Hn−i(X,R)→ R

induced by (α, β) 7→∫α ∧ β is a perfect pairing

8.3. THE HEAT EQUATION* 121

Proof. ∗ induces an isomorphism between the space of harmonic i-forms andn− i-forms. This proves directly that Hi(X,R) and Hn−i(X,R) are isomorphic.

Consider the mapλ : Hi(X,R)→ Hn−i(X,R)∗

given by λ(α) = β 7→∫α ∧ β. We need to prove that λ is an isomorphism. Since

these spaces have same dimension, it is enough to prove that ker(λ) = 0. But thisclear since λ(α)(∗α) 6= 0 whenever α is a nonzero harmonic form.

Exercises

1. Give a careful proof of lemma 8.2.1 and determine the signs.2. Show that an harmonic exact form dγ satisfies ||dγ||2 = 0. Use this to

prove the uniqueness in theorem 8.2.2.3. Prove corollary 8.2.6.4. Fill in the details of example 8.2.7.5. Let X be a compact Riemannian manifold with a group of isometries G,

i.e. G preserves the metric. Then show that if α is harmonic, then g∗α isalso harmonic for any g ∈ G.

6. Suppose that G is a connected Lie group with an action G × X → Xby isometries on compact Riemannian manifold X. Then show that anyharmonic form is G-invariant.

8.3. The Heat Equation*

We will outline an approach to 8.2.5 using the heat equation due Milgram andRosenbloom [MR]. However, we will deviate slightly from their presentation, whichis a bit too sketchy in places. The heuristic behind the proof is that if the formα is thought of as an initial temperature, then the temperature should approacha harmonic steady state as the manifold cools. So our task is to solve the heatequation:

∂A(t)∂t

= −∆A(t)(8.3.1)

A(0) = α(8.3.2)

for all t > 0, and study the behaviour as t→∞. For simplicial complexes, this wasdealt with the exercises of the first section.

We start with a few general remarks.

Lemma 8.3.1. If A(t) is a C∞ solution of (8.3.1), ||A(t)||2 is (nonstrictly)decreasing.

Proof. Upon differentiating ||A(t)||2 we obtain:

2〈∂A∂t,A〉 = −2〈∆A,A〉 = −2(||dA||2 + ||d∗A||2) ≤ 0.

Corollary 8.3.2. A solution to (8.3.1)-(8.3.2) would be unique.

Proof. Given two solutions, their difference satisfies A(0) = 0, so it remains0.


The starting point for the proof of existence is the observation that when Xis replaced by Euclidean space and α is a compactly supported function, then onehas an explicit solution to (8.3.1)-(8.3.2)

A(x, t) =∫

Rn

K(x, y, t)α(y)dy = 〈K(x, y, t), α(y)〉y

where the heat kernel

K(x, y, t) = (4πt)−n/2e−||x−y||2/4t

and the symbol 〈, 〉y is the L2 inner product with integration carried out withrespect to y.

This can be modified to make sense for a p-form on X by replacing K above by

K(x, y, t) = (4πt)−n/2e−||x−y||2/4t(

∑i

dxi ∧ dyi)p

When working with forms on X ×X × [0,∞), we will abuse notation a bit andwrite them as local expressions such as η(x, y, t). Then dxη(x, y, t) etcetera willindicate that the operations d . . . are preformed with y, t treated as constant (ormore correctly, these operations are preformed fibrewise along a projection).

Theorem 8.3.3. On any compact Riemannian manifold, there exists a C∞

p-form K(x, y, t) called the heat kernel such that for any p-form α,

A(x, t) = 〈K(x, y, t)α(y)〉ygives a solution to the heat equation (8.3.1) with A(x, 0) = α(x).

Proof. We sketch a proof due to Minakshisundaram when p = 0; a moredetail account can be found in [Ch]. This argument can be modified to give acomplete proof of the theorem, see [BGV, chap 2] or [P, sect 4]. Let δ(x, y)2 be anonnegative C∞ function on X×X which agrees with the square of the Riemanniandistance function in neighbourhood of the diagonal, and vanishes far away from it.Set

K0(x, y, t) = (4πt)−n/2e−δ(x,y)2/4t

This is only an approximation to the true heat kernel in general, but it does satisfya crucial property that it approaches the “δ-function of x−y” as t→ 0 in the sensethat

limt→0〈K0(x, y, t), α(y)〉y = α(x)

However, we will need to replace K0 by an approximation with better asymptoticproperties as t → 0. For each N ∈ N, Minakshisundaram-Pleijel have shown howto find C∞ functions ui(x, y) such that

K1(x, y, t) = K0(x, y, t)[u0(x, y) + tu1(x, y) + . . . tNuN (x, y)]

satisfies (∆x +

∂

∂t

)K1 = tNK0∆xuN ,

which can be written as tN−n/2e−δ(x,y)2/4t times a C∞-function. We want to em-phasize that the existence of the ui’s follows by direct calculation, and does notinvolve anything deep. Fix N > n/2 + 2, and choose K1 as above.


We define the convolution of two functions C(x, y, t) and B(x, y, t) by

(C ∗B)(x, y, t) =∫ t

0

〈C(x, z, t− τ), B(z, y, τ)〉zdτ

This is an associative operation. Let L =(∆x + ∂

∂t

)K1. Given a C∞ function

B(x, y, t), we claim that

(8.3.3)(

∆x +∂

∂t

)(K1 ∗B) = −B + L ∗B

Leth(x, y, t, τ) = 〈K1(x, z, t− τ), B(z, y, τ)〉z

be the integrand of K1 ∗B. Then (8.3.3) follows formally by adding

∂

∂t

∫ t

0

h(x, y, t, τ)dτ = limτ→t

h(x, y, t, τ) +∫ t

0

∂

∂th(x, y, t, τ)dτ

to

∆x

∫ t

0

h(x, y, t, τ)dτ =∫ t

0

∆xh(x, y, t, τ)dτ

Set

(8.3.4) K = K1 +K1 ∗ L+K1 ∗ L ∗ L+ . . . = K1 +K1 ∗ (L+ L ∗ L+ . . .)

We can obtain an estimate

|L∗`| ≤ tN+`−n/2−1C`

where C` is a constant which goes to 0 rapidly with `. This implies absolute uniformconvergence of this series on X2×[0, T ]. This also shows that K1∗(L+L∗L . . .)→ 0as t → 0, and therefore K also approaches the δ-function as t → 0. A similarestimate on the derivatives up to second order allows us apply (8.3.3) term by termto get a telescoping series(

∆x +∂

∂t

)K = L+ (−L+ L ∗ L) + (−L ∗ L+ L ∗ L ∗ L) + . . . = 0

It follows thatA(x, t) = 〈K(x, y, t), α(y)〉y

satisfies the heat equation and the required initial condition.

LetTt(α) = 〈K(x, y, t), α(y)〉y

with K as in the last theorem. This is the unique solution to (8.3.1) and (8.3.2).Theorem 8.2.5 can now be deduced from:

Theorem 8.3.4.(1) The semigroup property Tt1+t2 = Tt1Tt2 holds.(2) Tt is formally self adjoint.(3) Ttα converges to a C∞ harmonic form H(α).(4) The integral

G(α) =∫ ∞

0

(Ttα−Hα)dt

is well defined, and yields Green’s operator.


Proof. We give the main ideas. The semigroup property Tt1+t2 = Tt1Tt2holds because A(t1 + t2) can be obtained by solving the heat equation with initialcondition A(t2) and then evaluating it at t = t1.

To see that Tt is self adjoint, calculate∂

∂t〈Ttη, Tτξ〉 = 〈 ∂

∂tTtη, Tτξ〉 = −〈∆Ttη, Tτξ〉

= −〈Ttη,∆Tτξ〉 = 〈Ttη,∂

∂τTτξ〉 =

∂

∂τ〈Ttη, Tτξ〉

which implies that 〈Ttη, Tτξ〉 can be written as a function of t + τ , say g(t + τ).Therefore

〈Ttη, ξ〉 = g(t+ 0) = g(0 + t) = 〈η, Ttξ〉.Properties (1) and (2) imply that for h ≥ 0 we have

||Tt+2hα− Ttα||2 = ||Tt+2hα||2 + ||Ttα||2 − 2〈Tt+2hα, Ttα〉= ||Tt+2hα||2 + ||Ttα||2 − 2||Tt+hα||2

= (||Tt+2hα|| − ||Ttα||)2 − 2(||Tt+hα||2 − ||Tt+2hα|| · ||Ttα||)

||Ttα||2 converges thanks to lemma 8.3.1. Therefore ||Tt+2hα− Ttα||2 can be madearbitrarily small for large t. This implies that Ttα converges in the L2 sense to anL2 form H(α), i.e. an element of the Hilbert space completion of Ep(X). Fix τ > 0,the relations Ttα = TτTt−τα, implies in the limit that TτH(α) = H(α). Since, Tτis given by an integral transform with smooth kernel, it follows that H(α) is C∞.The equation TτH(α) = H(α) shows that H(α) is in fact harmonic. We also notethat H is formally self adjoint

〈Hα, β〉 = limt→∞〈Ttα, β〉 = lim

t→∞〈α, Ttβ〉 = 〈α,Hβ〉

The pointwise norms ||Ttα(x)−Hα(x)|| can be shown to decay rapidly enoughthat the integral

G(α) =∫ ∞

0

(Ttα−Hα)dt

is well defined. We will verify formally that this is Green’s operator:

∆G(α) =∫ ∞

0

∆Ttαdt = −∫ ∞

0

∂Ttα

∂tdt = α−H(α),

and for β harmonic

〈G(α), β〉 =∫ ∞

0

〈(Tt −H)α, β〉dt =∫ ∞

0

〈α, (Tt −H)β〉dt = 0

as required.

Let’s return to the example of a torus where things can be calculated explicitly.

Example 8.3.5. Let X = Rn/Zn. Given α as in (8.2.2), the solution to theheat equation with initial value α is given by

Ttα =∑λ∈Zn

∑aλ,i1...ipe

(2πiλ·x−4π2|λ|2t)dxi1 ∧ . . . ∧ dxip ,

and this converges to the harmonic projection

H(α) =∑

a0,i1...ipdxi1 ∧ . . . ∧ dxip


Ttα−Hα can be integrated term by term to obtain Green’s operator

G(α) =∑

λ∈Zn−0

∑ aλ,i1...ip4π2|λ|2

e2πiλ·xdxi1 ∧ . . . ∧ dxip .

Exercises

1. Suppose that α is an eigenfunction so that ∆α = λα. Give an explicitformula for Ttα.

2. Check that example 8.3.5 works as claimed.

CHAPTER 9

Toward Hodge theory for Complex Manifolds

In this chapter, we take the first few steps toward Hodge theory in the complexsetting. This is really a warm up for the next chapter.

9.1. Riemann Surfaces Revisited

In this section we tie up a loose end from chapter 6, by proving proposition6.4.7. Fix a compact Riemann surface X. In order to apply the techniques fromthe previous chapter, we need to choose a Riemannian metric which is a C∞ familyof inner products on the tangent spaces. But we need to impose a compatibilitycondition that multiplication by i =

√−1 preserves the angles determined by these

inner products. To say this more precisely, view X as two dimension real C∞

manifold. Choosing an analytic local coordinate z = x+ iy in a neighbourhood ofU , the vectors v1 = ∂

∂x and v2 = ∂∂y give a basis (or frame) of the real tangent sheaf

TX of X restricted to U . The automorphism Jp : TX |U → TX |U represented by(0 1−1 0

)in the basis v1, v2 is independent of this basis, and hence globally well defined.A Riemannian metric (, ) is said to be compatible with the complex structure, orHermitean, if the transformations Jp are orthogonal. In terms of the basis v1, v2

this forces the matrix of the bilinear form (, ) to be a positive multiple of I bysome function h. In coordinates, the metric would be represented by a tensorh(x, y)(dx⊗ dx+ dy⊗ dy). The volume form is represented by hdx∧ dy. It followsthat ∗dx = dy and ∗dy = −dx. In other words, ∗ is the transpose of J which isindependent of h. Once we have ∗, we can define all the operators from the lastchapter.

Standard partition of unity arguments show that Hermitean metrics alwaysexist. Fix one. (For our purposes one is as good as any other.)

Lemma 9.1.1. A 1-form is harmonic if and only if its (1, 0) and (0, 1) parts arerespectively holomorphic and antiholomorphic.

Proof. A 1-form α is harmonic if and only if dα = d ∗ α = 0. Given a localcoordinate z, it can be checked that ∗dz = −

√−1dz and ∗dz =

√−1dz. Therefore

the (1, 0) and (0, 1) parts of a harmonic 1-form α are closed. If α is a (1, 0)-form,then dα = ∂α. Thus α is closed if and only if it is holomorphic. Conjugation yieldsthe analogous statement for (0, 1)-forms.

Corollary 9.1.2. dim H0(X,Ω1X) equals the genus of X.

Proof. The genus is one half of the first Betti number dim H1(X,C). Bythe Hodge theorem this is the dimension of the space of harmonic 1-forms, which

127

128 9. TOWARD HODGE THEORY FOR COMPLEX MANIFOLDS

decomposes into a direct sum of the spaces of holomorphic and antiholomorphic1-forms. Both these spaces have the same dimension, since conjugation gives a realisomorphism between them.

Lemma 9.1.3. The image of ∆ and ∂∂ on E2(X) coincide.

Proof. On the space of 2-forms, we have ∆ = −d ∗ d∗. Computing in localcoordinates yields

∂∂f = −√−12

(∂2f

∂x2+∂2f

∂y2

)dx ∧ dy

and

d ∗ df =(∂2f

∂x2+∂2f

∂y2

)dx ∧ dy.

which implies the lemma.

Proposition 9.1.4. The map H1(X,OX) → H1(X,Ω1X) induced by d van-

ishes.

Proof. We use the descriptions of these spaces as ∂-cohomology groups 6.4.3.Given α ∈ E01(X), let β = dα. We have to show that β lies in the image of ∂.Applying theorem 8.2.5 we can write β = H(β) + ∆G(β). Since β is exact, wecan conclude that H(β) = 0 by corollary 8.2.6. Therefore β lies in the image of∂∂ = −∂∂.

Corollary 9.1.5. The map H1(X,C)→ H1(X,OX) is surjective, and dimH1(X,OX)coincides with the genus of X.

Proof. The surjectivity is immediate from the exact sequence

H1(X,C)→ H1(X,OX)→ H1(X,Ω1X)

The second part follows from the equation

dimH1(X,OX) = dimH1(X,C)− dimH0(X,ΩX)

Exercises

1. Show that H1(X,C) → H1(X,OX) can be identified the projection ofharmonic 1-forms to antiholomorphic (0, 1)-forms.

2. Calculate the spaces of harmonic and holomorphic one forms explicitly foran elliptic curve.

3. Show that the pairing (α, β) 7→∫Xα ∧ β is positive definite.

9.2. Dolbeault’s theorem

We now extend the results from Riemann surfaces to higher dimensions. Givenan n dimensional complex manifold X, let OX denote the sheaf of holomorphicfunctions. We can regard X as a 2n dimensional (real) C∞ manifold as explainedin section 2.2. As with Riemann surfaces, from now on C∞X will denote the sheaf ofcomplex valued C∞ functions. Likewise EkX will denote the sheaf of C∞ complexvalued k-forms. If EkX,R to denote the sheaf of real valued C∞ k-forms as definedin section 2.6. Then

EkX(U) = C⊗R EkX,R(U).

9.2. DOLBEAULT’S THEOREM 129

By a real structure on a complex vector space V , we mean real vector space VR andan isomorphism C⊗VR ∼= V . This gives rise to a C-antilinear involution v 7→ v givenby a⊗ v = a ⊗ v. Conversely, such an involution gives rise to the real structureVR = v | v = v. In particular, EkX(U) has a natural real structure.

The sheaf of holomorphic p-forms ΩpX is a subsheaf of EpX stable under mul-tiplication by OX . This sheaf is locally free as an OX -module. If z1, . . . zn areholomorphic coordinates defined on an open set U ⊂ X,

dzi1 ∧ . . . ∧ dzip | i1 < . . . < ip

gives a basis for ΩpX(U). To simplify our formulas, we will write the above expres-sions as dzI where I = i1, . . . , ip.

Definition 9.2.1. Let E(p,0)X denote the C∞ submodule of EpX generated by ΩpX .

Let E(0,p)X = E(p,0)

X , and E(p,q)X = E(p,0)

X ∧ E(0,q)X .

In local coordinates dzI∧dzJ | #I = p,#J = q, gives a basis of E(p,q)X (U). All

of the operations of sections 6.3 and 6.4 can be extended to the higher dimensionalcase. The operators

∂ : E(p,q)X → E(p+1,q)

X

and

∂ : E(p,q)X → E(p,q+1)

X

are given locally by

∂(∑I,J

fI,JdzI ∧ dzJ) =∑I,J

n∑i=1

∂fI,J∂zi

dzi ∧ dzI ∧ dzJ

∂(∑I,J

fI,JdzI ∧ dzJ) =∑I,J

n∑j=1

∂fI,J∂zj

dzj ∧ dzI ∧ dzJ

The identities

d = ∂ + ∂(9.2.1)∂2 = ∂2 = 0∂∂ + ∂∂ = 0

hold.The analogue of theorem 6.3.2 is

Theorem 9.2.2. Let D ⊂ Cn be an open polydisk (i.e. a product of disks).Given α ∈ E(p,q)(D) with ∂α = 0, there exists β ∈ E(p,q−1)(D) such that α = ∂β

Proof. See [GH, pp. 25-26].

Corollary 9.2.3 (Dolbeault’s Theorem).

0→ ΩpX → E(p,0)X

∂−→ E(p,1)X . . .

is a fine resolution.


Proof. Given a (p, 0) form∑I fIdzI ,

∂(∑I

fIdzI) =∑I

n∑j=1

∂fI∂zj

dzj ∧ dzI = 0

if and only it is holomorphic. In other words, ΩpX is the kernel of the ∂ operatoron E(p,0)

X , and the corollary follows.

Exercises

1. Check the identities 9.2.1 when dimX = 2.

9.3. Complex Tori

A complex torus is a quotient X = V/L of a finite dimensional complex vectorspace by a lattice (i.e. a discrete subgroup of maximal rank). Thus it is both acomplex manifold and a torus. Let us identify V with Cn. Let z1, . . . zn be thestandard complex coordinates on Cn, and let xi = Re(zi), yi = Im(zi).

We have already seen that harmonic forms with respect to the flat, i.e. Eu-clidean, metric are the forms with constant coefficients. The space of harmonicforms can be decomposed into (p, q) type.

H(p,q) =⊕

#I=p,#J=q

CdzI ∧ dzJ

These forms are certainly ∂-closed, and hence define ∂-cohomology classes. In factthey give a basis for cohomology.

Proposition 9.3.1. H(p,q) ∼= Hq(X,ΩpX).

Corollary 9.3.2. Hq(X,ΩpX) ∼= ∧pCn ⊗ ∧qCn

The isomorphism in the corollary is highly noncanonical. A more natural iden-tification is

Hq(X,ΩpX) ∼= ∧pV ∗ ⊗ ∧qV ∗

where V ∗ is the usual dual, and V ∗ is the set of antilinear maps from V to C.The proof of proposition 9.3.1 hinges on a certain identity between Laplacians

that we now define. The space of forms carry inner products as in section 8.2 whereX is equipped with the flat metric. Let ∂∗ and ∂∗ denote the adjoints to ∂ and ∂respectively. These will be calculated explicitly below. We can define the ∂ and∂-Laplacians by

∆∂ = ∂∗∂ + ∂∂∗,

∆∂ = ∂∗∂ + ∂∂∗

Lemma 9.3.3. ∆ = 2∆∂ = 2∆∂ .

This can be checked directly by an explicit calculation of the Laplacians [GH,p. 106]. However, we will sketch a more complicated proof, since it this argumentwhich generalizes. We introduce a number of auxiliary operators. Let ik andik denote contraction with the vector fields 2 ∂

∂zkand 2 ∂

∂zk. Thus for example

ik(dzk ∧ α) = 2α. If we choose our Euclidean metric so that monomials in dxi, dyj

9.3. COMPLEX TORI 131

are orthonormal, then the contractions ik and ik can be checked to be adjoints todzk∧ and dzk∧. Let

ω =√−12

∑dzk ∧ dzk =

∑dxk ∧ dyk

Lα = ω ∧ αand

Λ = −√−12

∑ikik

The operators L and Λ are adjoint. Using integration by parts (see [GH, p. 113]),we get explicit formulas

∂∗α = −∑ ∂

∂zkikα

∂∗α = −∑ ∂

∂zkikα

where the derivatives above are taken coefficient-wise.Let [A,B] = AB − BA denote the commutator. Then we have the first order

Kahler identities:

Proposition 9.3.4.1. [Λ, ∂] = −

√−1∂∗

2. [Λ, ∂] =√−1∂∗

Proof. We check the second identity on the space of (1, 1) forms. The generalcase is more involved notationally but not essentially harder, see [GH, p. 114].There are two cases. First suppose that α = fdzj ∧ dzk, with j 6= k. Then

[Λ, ∂]α = Λ∂α

= Λ

(∑m

∂f

∂zmdzm ∧ dzj ∧ dzk

)

= 2√−1

∂f

∂zkdzj

=√−1∂∗α

Next suppose that α = fdzk ∧ dzk. Then

[Λ, ∂]α = Λ

∑m 6=k

∂f

∂zmdzm ∧ dzk ∧ dzk

− ∂ (−2√−1f

)

= −2√−1

∑m6=k

∂f

∂zmdzm −

∑m

∂f

∂zmdzm

= 2

√−1

∂f

∂zkdzk

=√−1∂∗α

Upon substituting these identities into the definitions of the various Laplacianssome remarkable cancellations take place:


Proof of lemma 9.3.3. We first establish ∂∂∗ + ∂∗∂ = 0,√−1(∂∂∗ + ∂∗∂) = ∂(Λ∂ − ∂Λ) + (Λ∂ − ∂Λ)∂

= ∂Λ∂ − ∂Λ∂ = 0

Similarly, we have ∂∗∂ + ∂∂∗ = 0.Next expand ∆,

∆ = (∂ + ∂)(∂∗ + ∂∗) + (∂∗ + ∂∗)(∂ + ∂)= (∂∂∗ + ∂∗∂) + (∂∂∗ + ∂∗∂) + (∂∂∗ + ∂∗∂) + (∂∗∂ + ∂∂∗)= (∂∂∗ + ∂∗∂) + (∂∂∗ + ∂∗∂) + (∂∂∗ + ∂∗∂) + (∂∗∂ + ∂∂∗)= ∆∂ + ∆∂

Finally, we check ∆∂ = ∆∂ ,

−√−1∆∂ = ∂(Λ∂ − ∂Λ) + (Λ∂ − ∂Λ)∂

= ∂Λ∂ − ∂∂Λ + Λ∂∂ − ∂Λ∂= (∂Λ− Λ∂)∂ + ∂(∂Λ− Λ∂)

= −√−1∆∂

Proof of proposition 9.3.1. By Dolbeault’s theorem

Hq(X,ΩpX) ∼=ker[∂ : E(p,q)(X)→ E(p,q+1)]im[∂ : E(p,q−1)(X)→ E(p,q)]

Let α be a ∂-closed (p, q)-form. Decompose

α = β + ∆γ = β + 2∆∂γ = β + ∂γ1 + ∂∗γ2

with β harmonic, which possible by theorem 8.2.5. We have

||∂∗γ2||2 = 〈γ2, ∂∂∗γ2〉 = 〈γ2, ∂α〉 = 0.

It is left as an exercise to check that β is of type (p, q), and that it is unique Thereforethe ∂-class of α has a unique representative by a constant (p, q)-form.

Exercises

1. Check the first order Kahler identities 9.3.4 on space of all 2-forms.2. Show that β + ∂γ = 0 forces β = 0 if β is harmonic.3. Suppose that α = β + ∂γ and that α is of type (p, q) and β harmonic. By

decomposing β =∑

β(p′,q′) and γ =∑

γ(p′,q′) into (p′, q′) type and usingthe previous exercise, prove that β is of type (p, q) and unique.

CHAPTER 10

Kahler manifolds

In this chapter, we extend the results of the previous chapter to an importantclass of manifolds called Kahler manifolds. These are complex manifolds which aspecial metric called a Kahler metric. Unlike Riemannian or Hermitean metrics,there are topological obstructions for a manifold to carry a Kahler metric. Fortu-nately, many of the examples of interest coming from algebraic geometry do possesssuch metrics.

10.1. Kahler metrics

Let X be a compact complex manifold with complex dimension n. A Hermiteanmetric H on X could be defined as in the previous chapter as a Riemannian metricfor which multiplication by

√−1 is orthogonal. However, it is more convenient for

our purposes to view it as a choice of Hermitean inner product on the complextangent spaces which vary in C∞ fashion. More precisely, H would be given by asection of E(1,0)

X ⊗ E(0,1)X , such that in some (any) locally coordinate system zi =

xi +√−1yi around each point, H is given by

H =∑

hijdzi ⊗ dzj

with hij positive definite Hermitean. The real part of this matrix is positive definitesymmetric, and the tensor∑

Re(hij)(dxi ⊗ dxi + dyi ⊗ dyi)

gives a globally defined Riemannian structure on X. We also have a (1, 1)-form ω

called the Kahler form which is the normalized image of H under E(1,0)X ⊗ E(0,1)

X →E(1,0)X ∧ E(0,1)

X . In coordinates

ω =√−12

∑hijdzi ∧ dzj .

The normalization makes ω real, i.e. ω = ω. It is clear from this formula, that ωdetermines the metric.

Definition 10.1.1. A Hermitean metric on X is called a Kahler metric forany p ∈ X if there exist analytic coordinates z1, . . . zn with zi = 0 at p, for whichthe metric becomes Euclidean up to second order:

hij ≡ δij mod (z1, . . . zn)2

A Kahler manifold is a complex manifold which admits a Kahler metric. (Some-times the term is used for a manifold with a fixed Kahler metric.)

133

134 10. KAHLER MANIFOLDS

In such a coordinate system, a Taylor expansion gives

ω =√−12

∑dzi ∧ dzi + terms of 2nd order and higher

Therefore dω = 0 at zi = 0. Since such coordinates can be chosen around point,dω is identically zero. This gives a nontrivial obstruction for a Hermitean metricto be Kahler. In fact, this condition characterizes Kahler metrics and often takenas the definition:

Proposition 10.1.2. Given a Hermitean metric H, the following are equivalent(1) H is Kahler.(2) The Kahler form is closed: dω = 0.(3) The Kahler form is locally expressible as ω = ∂∂f .

Proof. That (1) implies (2) was explained above. See [GH, p 107] for theconverse. (3) clearly implies (2), The converse will be proved in the exercises.

We will refer the cohomology class of ω as the Kahler class. The function f suchthat ω =

√−1∂∂f is called a Kahler potential. A function f is plurisubharmonic

if it is a Kahler potential, or equivalently in coordinates this means∑ ∂2f

∂zi∂zjξiξj > 0

for any nonzero vector ξ.Basic examples of compact Kahler manifolds are:

Example 10.1.3. Any Hermitean metric on a Riemann surface is Kahler sincedω vanishes for trivial reasons.

Example 10.1.4. Complex tori are Kahler. Any flat metric will do.

Example 10.1.5. The Fubini-Study metric on Pn is the unique Hermitean met-ric with Kahler form which pulls back to

√−1

2π∂∂ log(|z0|2 + . . .+ |zn|2)

under the canonical map Cn+1 − 0 → Pn. In inhomogeneous coordinates we canwrite ω as √

−12π

∂∂ log

(∣∣∣∣z0

zi

∣∣∣∣2 + . . .+ 1 + . . .+∣∣∣∣znzi∣∣∣∣2)

This form is obviously closed. Since H2(Pn,C) is one dimensional, the Kahler class[ω] would have to be a nonzero multiple of c1(O(1)). In fact, the constants in theformulas are chosen so that these coincide.

Definition 10.1.6. A line bundle L on a compact complex manifold X is calledvery ample if there is an embedding X ⊂ Pn such that L ∼= OPn(1)|X . L is ampleif some positive tensor power L⊗ L⊗ . . . is very ample.

The key examples of Kahler manifolds are provided by

Lemma 10.1.7. A complex submanifold of a Kahler manifold inherits a Kahlermetric such that the Kahler class is the restriction of the Kahler class of the ambientmanifold.

10.2. THE HODGE DECOMPOSITION 135

Proof. The Kahler form locally has a plurisubharmonic potential f . It followsimmediately from the definition, that f restricts to a plurisubharmonic function oncomplex submanifold. Thus the Kahler form will restrict to a Kahler form.

Corollary 10.1.8. A smooth projective variety has a Kahler metric, withKahler class equal to c1(L) for L very ample.

Since c1(L⊗n) = nc1(L), we can replace “very ample” by “ample” above.

Exercises

1. Prove 2 ⇒ 3 in proposition 10.1.2. (Use Poincare’s lemma to locally solveω = dα, then apply the ∂-Poincare lemma (9.2.2) to the (0, 1)-part of αand the conjugate of the (1, 0)-part.)

2. Show that the Fubini-Study metric is Kahler by checking that∣∣∣∣z0

zi

∣∣∣∣2 + . . .+ 1 + . . .+∣∣∣∣znzi∣∣∣∣2

is plurisubharmonic.3. Show that the product of two Kahler manifolds is Kahler.

10.2. The Hodge decomposition

Fix an n dimensional Kahler manifold X with a Kahler metric. Since Kahlermetrics are Euclidean up to second order, we have the following metatheorem:Any identity involving geometrically defined first order differential operators on Eu-clidean space will automatically extend to Kahler manifolds. Let us introduce therelevant operators. Since X is a complex manifold, it has a canonical orientationgiven by declaring ∂

∂x1, ∂∂y1

, ∂∂x2

, . . . to be positively oriented for any choice of ana-lytic local coordinates. Thus the Hodge star operator associated to the Riemannianstructure can be defined. The operator ∗ will be extended to a C-linear operatoron E•X , and set1 ∗(α) = ∗α = ∗α. If we let ( , ) be the pointwise Hermitean innerproduct on E•(X), then (8.2.1) becomes

α ∧ ∗β = (α, β)dvol.

It follows easily from this that

∗E(p,q)(X) ⊆ E(n−p,n−q)(X)

∗E(p,q)(X) ⊆ E(n−q,n−p)(X).Let ∂∗ = −∗∂∗ = −∗∂∗ and ∂∗ = −∗∂∗ = −∗ ∂∗. These operators are the adjointsof ∂ and ∂, and have bidegree (−1, 0) and (0,−1) respectively. Then we can definethe operators with bidegrees indicated on the right:

∆∂ = ∂∗∂ + ∂∂∗ (0, 0)∆∂ = ∂∗∂ + ∂∂∗ (0, 0)L = ω∧ (1, 1)Λ = − ∗ L∗ = −∗L∗ (−1,−1)

A form is called ∂-harmonic if it lies in the kernel of ∆∂ . For a general Hermiteanmanifold there is no relation between harmonicity and ∂-harmonicity. However,these notions do coincide for the class of Kahler manifolds, by the following:

1Many authors, notably Griffiths and Harris [GH], would define ∗ to be what we call ∗.


Theorem 10.2.1. ∆ = 2∆∂ = 2∆∂ .

Proof. Since Laplacians are second order, the above metatheorem cannot beapplied directly. However, the first order identities of proposition 9.3.4 do generalizeto X by this principle. The rest of the argument is the same the proof of lemma9.3.3.

Corollary 10.2.2. If X is compact then Hq(X,ΩpX) is isomorphic to the spaceof harmonic (p, q)-forms.

Proof. Since harmonic forms are ∂-closed (exercise), we have a map from thespace of harmonic (p, q)-forms to Hq(X,ΩpX). The rest of the argument is identicalto the proof of proposition 9.3.1.

We obtain the following special case of Serre duality as a consequence:

Corollary 10.2.3. When X is compact, Hp(X,ΩqX) ∼= Hn−p(X,Ωn−qX )

Proof. ∗ induces an R-linear isomorphism between the corresponding spacesof harmonic forms.

Theorem 10.2.4 (The Hodge decomposition). Suppose that X is a compactKahler manifold. Then a differential form is harmonic if and only if its (p, q) com-ponents are. Consequently we have (for the moment) noncanonical isomorphisms

Hi(X,C) ∼=⊕p+q=i

Hq(X,ΩpX).

Furthermore, complex conjugation induces R-linear isomorphisms between the spaceof harmonic (p, q) and (q, p) forms. Therefore

Hq(X,ΩpX) ∼= Hp(X,ΩqX).

Proof. The operator ∆∂ preserves the decomposition

E i(X) =⊕p+q=i

E(p,q)(X)

Therefore a form is harmonic if and only if its (p, q) components are. Since com-plex conjugation commutes with ∆, conjugation preserves harmonicity. These ar-guments, together with corollary 10.2.2, finishes the proof.

Corollary 10.2.5. The Hodge numbers hpq(X) = dimHq(X,ΩpX) are finitedimensional.

This theorem has a number of implications for the topology of compact Kahlermanifolds.

Corollary 10.2.6. If i is odd then the ith Betti number bi of X is even.

Proof.bi = 2

∑p<q

hpq

Corollary 10.2.7 (Johnson-Rees). The fundamental group of X cannot befree (unless its trivial).

10.3. PICARD GROUPS 137

Proof. The proof is taken from [A] which gives a bit more detail. If π1(X)were free on an odd number of generators, then the first Betti number would beodd which is ruled out as above. If π1(X) is free on an even number of generators,then it can be checked that it contains a subgroup of finite index which is free onan odd number of generators. This subgroup is the fundamental group of a finitesheeted covering Y → X. Since Y inherits a Kahler structure from X, this is againimpossible.

The finite dimensionality of Hq(X,ΩpX) is also true for compact complex nonKahler manifolds by theorem 15.2.10, however the other corollaries may fail. TheHodge numbers gives an important set of holomorphic invariants for X. We canvisualize them by arranging them in a diamond:

h00 = 1h10 h01

h20 h11 h02

. . .hn,n−1 hn−1,n

hnn = 1

The previous results implies that this picture has both vertical and lateralsymmetry (e.g h10 = h01 = hn,n−1 = hn−1,n).

Exercises

1. Prove that a form α on a compact Kahler manifold is harmonic if and onlyif ∂α = ∂∗α = 0.

2. The Hopf surface S is complex manifold obtained as a quotient of C2−0by Z acting by z 7→ 2nz. Show that S is homeomorphic to S1 × S3, andconclude that it cannot be Kahler.

3. Using the metatheorem, verify that dvol = ω∧ω2 on a 2 dimensional Kahler

manifold. Conclude that [ω] 6= 0 and hence that b2 6= 0.4. Generalize the previous exercise to all dimensions, and conclude that the

even Betti numbers of a compact Kahler manifold are all nonzero.

10.3. Picard groups

Let us write H2(X,Z) ∩ H11(X) for the preimage of H11(X) under the mapH2(X,Z)→ H2(X,C). Note that this contains the torsion subgroup H2(X,Z)tors.

Theorem 10.3.1. Let X be compact Kahler manifold. Then Pic(X) fits intoan exact sequence

0→ Pic0(X)→ Pic(X)→ H2(X,Z) ∩H11(X)→ 0

with Pic0(X) a complex torus of dimension h01(X)

There are two assertions which will be proved separately.

Proposition 10.3.2. Pic0(X) = ker(c1) is a complex torus.

Proof. From the exponential sequence, we obtain

H1(X,Z)→ H1(X,OX)→ Pic(X)→ H2(X,Z)


Pic0 is the cokernel of the first map, that is H1(X,OX)/H1(X,Z). We have toprove that H1(X,Z) sits inside H1(X,OX) as a lattice. It suffices to prove thatthe natural map

π : H1(X,R)→ H1(X,OX)is an isomorphism of real vector spaces. We know that that these space have thesame real dimension b1 = 2h10, so it is enough to check that π is injective. TheHodge decomposition implies that α ∈ H1(X,R) can be represented by a sum ofa harmonic (1, 0)-form α1 and a harmonic (0, 1)-form α2. Since α = α, α1 = α2.π(α) is just α1. Therefore π(α) = 0 implies that α = 0.

Theorem 10.3.3 (The Lefschetz (1, 1) theorem). c1(Pic(X)) = H2(X,Z) ∩H11(X)

Proof. The map f : H2(X,Z)→ H2(X,OX) can be factored as H2(X,Z)→H2(X,C) followed by projection of the space of harmonic 2-forms to the space ofharmonic (2, 0)-forms. From the exponential sequence c1(Pic(X)) = ker(f), andthis certainly contains H2(X,Z) ∩H11(X).

Conversely suppose that α ∈ ker(f), then its image in H2(X,C) can be repre-sented by a sum of a harmonic (0, 2)-form α1 and a harmonic (1, 1)-form α2. Sinceα = α, α1 must be zero. Therefore α ∈ H2(X,Z) ∩H11(X).

Pic0(X) is called the Picard torus (or variety when X is projective). When Xis a compact Riemann surface, Pic0(X) is usually called the Jacobian and denotedby J(X)

Example 10.3.4. When X = V/L is a complex torus, Pic0(X) is a new toruscalled the dual torus. Using corollary 9.3.2, it can be seen to be isomorphic toV ∗/L∗, where V ∗ is the antilinear dual and

L∗ = λ ∈ V ∗ |, Im(λ)(L) ⊆ Z

The dual of the Picard torus is called the Albanese torus Alb(X). Since H01 =H01, Alb(X) is isomorphic to

H0(X,Ω1X)∗

H1(X,Z)/H1(X,Z)tors,

where the elements γ of the denominator are identified with the functionals α 7→∫γα in numerator.

Exercises

1. Let X be a compact Riemann surface. Show that the pairing < α, β >=∫Xα ∧ β induces an isomorphism J(X) ∼= Alb(X); in other words J(X) is

self dual.2. Fix a base point x0 ∈ X. Show that the so called Abel-Jacobi map α : X →Alb(X) given by x 7→

∫ xx0

(which is well defined modulo H1) is holomorphic.3. Show that every holomorphic 1-form on X is the pullback of a holomorphic

1-form from Alb(X). In particular, the Abel-Jacobi map cannot be constantif h01 6= 0.

4. Consider the map Xn → Alb(X) given by (x1, . . . xn) 7→∑α(xi). Show

that this is surjective for some n. (Hint: calculate the derivate at a generalpoint.)

CHAPTER 11

Homological methods in Hodge theory

In this chapter, we abstract things by introducing the language of Hodge struc-tures. This along with some tools from homological algebra will allow us to obtain acanonical Hodge decomposition on a compact Kahler manifold. We can also extendthis to certain non Kahler manifolds.

11.1. Pure Hodge structures

It is to useful isolate the purely linear algebraic features of the Hodge decom-position. We define a pure real Hodge structure of weight m to be finite dimensionalcomplex vector space with a real structure HR, and a bigrading

H =⊕

p+q=m

Hpq

satisfying Hpq = Hqp. We generally use the same symbol for Hodge structure andthe underlying vector space. A (pure weight m) Hodge structure is a real Hodgestructure H together with a choice of a finitely generated abelian group HZ and anisomorphism HZ ⊗ R ∼= HR. Even though the abelian group HZ may have torsion,it is helpful to think of it as “lattice” in HR. Rational Hodge structures are definedin a similar way. Given a pure Hodge structure, define the Hodge filtration by

F pHC =⊕p′≥p

Hpq.

The following is elementary.

Lemma 11.1.1. If H is a pure Hodge structure of weight m then

HC = F p ⊕ Fm−p+1

for all p. Conversely if F • is a descending filtration satisfying F a = HC and F b = 0for some a, b ∈ Z and satisfying the above identity, then

Hpq = F p ∩ F q

defines a pure Hodge structure of weight m.

The most natural examples of Hodge structures come compact Kahler mani-folds: if HZ = Hi(X,Z) with the Hodge decomposition on Hi(X,C) ∼= HZ ⊗ C,then we get a Hodge structure of weight i. It is easy to manufacture other examples.For every integer i, there is a rank one Hodge structure Z(i) of weight −2i. Herethe underlying space is C, with H(−i,−i) = C, and lattice HZ = (2π

√−1)iZ (these

factors should be ignored on first reading). The collection of Hodge structuresforms a category HS, where a morphism is linear map f preserving the lattices andthe bigradings. In particular, morphisms between Hodge structure with differentweights must vanish. This category has the following operations: direct sums of

139

140 11. HOMOLOGICAL METHODS IN HODGE THEORY

Hodge structures of the same weight (we will eventually relax this), and (unre-stricted) tensor products and duals. Explicitly, given Hodge structures H and Gof weights n and m, their tensor product H ⊗Z G is equipped with a weight n+mHodge structure with bigrading

(H ⊗G)pq =⊕

p′+p′′=p,q′+q′′=q

Hp′q′ ⊗Gp′′q′′ .

If m = n, their direct sum H ⊕G is equipped with the weight m Hodge structure

(H ⊕G)pq =⊕

p+q=m

Hpq ⊕Gpq.

The dual H∗ = Hom(H,Z) is equipped with a weight −n Hodge structure withbigrading

(H∗)pq = (H−p,−q)∗.

The operation H 7→ H(i) is called the Tate twist. It has the effect of leaving Hunchanged and shifting the bigrading by (−i,−i).

The obvious isomorphism invariants of a Hodge structure H are its Hodgenumbers dimHpq. However, this the doesn’t completely characterize them. In fact,there are usually uncountably many Hodge structures with a given set of Hodgenumbers. In some cases, these Hodge structures reflect the underlying geometry.For example.

Lemma 11.1.2. Consider, the set of 2g dimensional Hodge structures H ofweight 1 with dimH10 = dimH01 = g. There is a one to one correspondence be-tween isomorphism classes of Hodge structures as above and g dimensional complextori given by

H 7→ HC

HZ + F 1

Exercises

1. Show that there are no free rank one pure Hodge structures of odd weight,and up to isomorphism a unique rank one Hodge structure for every evenweight.

2. Show that HomHS(Z(0), H∗ ⊗G) ∼= HomHS(H,G), where3. Prove lemma 11.1.1.4. Prove lemma 11.1.2.

11.2. Canonical Hodge Decomposition

The Hodge decomposition involved harmonic forms so it is tied up with theKahler metric. It is possible to reformulate it, so as to make it independent of thechoice of metric. Let us see how this works for a compact Riemann surface X. Wehave an exact sequence

0→ CX → OX → Ω1X → 0

and we saw in lemma 6.4.6 that the induced map

H0(X,Ω1X)→ H1(X,C)

11.2. CANONICAL HODGE DECOMPOSITION 141

is injective. If we define

F 0H1(X,C) = H1(X,C)F 1H1(X,C) = im[H0(X,Ω1

X)→ H1(X,C)]F 2H1(X,C) = 0

then this together with the isomorphism H1(X,C) = H1(X,Z) ⊗ C determines apure Hodge structure of weight 1. To see this choose a metric which is automaticallyKahler because dimX = 1. Then H1(X,C) is isomorphic to a direct sum of thespace of harmonic (1, 0)-forms which maps to F 1 and the space of harmonic (0, 1)forms which maps to F 1.

Before proceeding with the higher dimensional version, we need some facts fromhomological algebra. Let

C• = . . . Cad→ Ca+1 → . . .

be a complex of vectors spaces (or modules or...). It is convenient to allow theindices to vary over Z, but we will require that it is bounded below which meansthat Ca = 0 for all a 0. Let us suppose that each Ci is equipped with a filtrationF pCi ⊇ F p+1Ci . . . which is preserved by d, i.e. dF pCi ⊆ F pCi+1. This impliesthat each F pC• is a subcomplex. We suppose further that F • biregular , whichmeans that for each i there exists a and b with F aCi = Ci and F bCi = 0. We geta map on cohomology

φp : H•(F pC•)→ H•(C•)and we let F pH•(C•) be the image. Define GrpHi(C•) = F pHi(C•)/F p+1Hi(C•).When C• is a complex of vector spaces, there are noncanonical isomorphisms

Hi(C•) =⊕p

GrpHi(C•).

The filtration is said to be strictly compatible with differentials of C•, or simply juststrict if all the φ’s are injective. Let GrpFC

• = GrpC• = F pC•/F p+1C•. Then wehave a short exact sequence of complexes:

0→ F p+1C → F pC → GrpC → 0

From which we get a connecting map δ : Hi(GrpC•) → Hi+1(F p+1C•). This canbe described explicitly as follows. Given x ∈ Hi(GrpC•), it can be lifted to anelement x ∈ F pCi such that dx ∈ F p+1Ci+1. Then δ(x) is represented by dx.

Proposition 11.2.1. The following are equivalent(1) F is strict.(2) F pCi+1 ∩ dCi = dF pCi for all i and p.(3) The connecting maps δ : Hi(GrpC•)→ Hi+1(F p+1C•) vanish for all i and

p.

proof (Su-Jeong Kang). (1)⇒ (2). Suppose that z ∈ F pCi+1 ∩ dCi. Thenz = dx ∈ F pCi+1 for some x ∈ Ci. Thus we have z ∈ ker[d : F pCi+1 → F pCi+2].Let z ∈ Hi+1(F pC•) denote the cohomology class of z. Note that φp(z) = 0, sincez = dx. Hence from the assumption that F is strict, z = 0 in Hi+1(F pC•) orequivalently z = dy for some y ∈ F pCi. This shows F pCi+1 ∩ dCi ⊆ dF pCi. Thereverse inclusion is clear.


(2) ⇒ (3). Let x ∈ Hi(GrpC•). This lifts to an element x ∈ F pCi withdx ∈ F p+1Ci+1 as above. Then from the assumption (2), we have

dx ∈ F p+1Ci+1 ∩ dCi = dF p+1Ci

Since δ(x) is represented by dx, we have δ(x) = 0 in Hi+1(F p+1C•).(3) ⇒ (1). For each i, φp can be expressed as finite a composition

Hi(F pC•)→ Hi(F p−1C•)→ Hi(F p−2C•) . . .

These maps are all injective by assumption, since their kernels would be the imagesof the connecting maps.

Corollary 11.2.2. GrpHi(C•) is a subquotient of Hi(GrpC•), which meansthat there is a diagram

Hi(GrpC•) ⊇ Ii,p → GrpHi(C•)

where the last map is onto. Isomorphisms GrpHi(C•) ∼= Ii,p ∼= Hi(GrpC•) hold,for all i, p, if and only if F is strict.

Proof. Let Ii,p = im[Hi(F p) → Hi(GrpC•)] then the surjection Hi(F p) →GrpHi(C•) factors through I. The remaining statement follows from (3) and adiagram chase.

Corollary 11.2.3. Suppose that C• is a complex of vector spaces over a fieldsuch that dimHi(GrpC•) <∞ for all i, p. Then

dimHi(C•) ≤∑p

dimHi(GrpC•),

and equality holds for all i if and only if F is strict. In which case, we also have

dimF pHi(C•) =∑p′≥p

dimHi(Grp′C•).

Proof. We have

dimHi(C•) =∑p

dimGrpHi(C•) ≤∑p

dim Iip ≤∑p

dimHi(GrpC•)

and equality is equivalent to strictness of F by the previous corollary. The laststatement is left as an exercise.

These results are usually formulated in terms of spectral sequences which wehave chosen to avoid. In this language the last corollary says that F is strict ifand only if the associated spectral sequence degenerates at E1. This is partiallyexplained in the exercises.

Let X be a complex manifold, then the de Rham complex E•(X) has a filtrationcalled the Hodge filtration:

F pE•(X) =∑p′≥p

Ep′q(X)

Its conjugate equalsF qE•(X) =

∑q′≥q

Epq′(X)

11.2. CANONICAL HODGE DECOMPOSITION 143

Theorem 11.2.4. If X is compact Kahler, the Hodge filtration is strict. Theassociated filtration F •Hi(X,C) gives a Hodge structure

Hi(X,Z)⊗ C ∼= Hi(X,C) =⊕p+q=i

Hpq(X)

of weight i, where

Hpq(X) = F pHi(X,C) ∩ F qHi(X,C) ∼= Hq(X,ΩpX).

Proof. Dolbeault’s theorem implies that Hq(GrpE•(X)) = Hq(E(p,•)X (X)) is

isomorphic to Hq(X,ΩpX). Therefore F is strict by corollary 11.2.3 and the Hodgedecomposition. By conjugation, we see that F is also strict. Furthermore, thesefacts together with corollary 11.2.3 gives

(11.2.1) dimF pHi(X,C) = hp,i−p(X) + hp+1,i−p−1(X) + . . .

and

(11.2.2) dim F i−p+1Hi(X,C) = hp−1,i−p+1(X) + hp−2,i−p+2(X) + . . .

A cohomology class lies in F pHi(X,C) (respectively F i−p+1Hi(X,C)) if and onlyif it can be represented by a form in F pE•(X) (respectively F i−p+1E•(X)). ThusHi(X,C) is the sum of these subspaces. Using (11.2.1) and (11.2.2), we see thatit is a direct sum. Therefore the filtrations determine a pure Hodge structure ofweight i on Hi(X,C).

Even though harmonic theory is needed to verify that this Hodge structure.It should be clear that it involves only the holomorphic structure and not involvethe metric. Thus we have obtained a canonical Hodge decomposition. Canonical isreally synonymous to functorial:

Corollary 11.2.5. If f : X → Y is a holomorphic map of compact Kahlermanifolds, then the pullback map f∗ : Hi(Y,Z) → Hi(X,Z) is compatible with theHodge structures.

Corollary 11.2.6. Global holomorphic differential forms on X are closed.

Proof. Strictness implies that the maps

d : H0(X,ΩpX)→ H0(X,Ωp+1X )

vanish by the exercises.

This corollary, and hence the theorem, can fail for compact complex non Kahlermanifolds. See the exercises.

Theorem 11.2.7. If X is a compact Kahler manifold, the cup product

Hi(X)⊗Hj(X)→ Hi+j(X)

is a morphism of Hodge structures.

The proof comes down to the observation that

F pE• ∧ F qE• ⊆ F p+qE•

For the corollaries, we work with rational Hodge structures. We have compat-ibility with Poincare duality:


Corollary 11.2.8. If dimX = n, then Poincare duality gives an isomorphismof Hodge structures

Hi(X) ∼= [H2n−i(X)∗](−n)

We have compatibility with the Kunneth formula:

Corollary 11.2.9. If X and Y are compact Kahler manifolds, then⊕i+j=k

Hi(X)⊗Hj(Y ) ∼= Hk(X × Y )

is an isomorphism of Hodge structures.

We have compatibility with Gysin map:

Corollary 11.2.10. If f : X → Y is a holomorphic map of compact Kahlermanifolds of dimension n and m respectively, the Gysin map is a morphism

Hi(X)→ Hi+2(m−n)(Y )(n−m)

Exercises

1. Finish the proof of 11.2.3.2. Let C•, F • be as above, and define Epq1 = Hp+q(GrpC•) and d1 : Epq1 →Ep+1,q

1 as the connecting map associated to

0→ GrpC• → F pC•/F p+2C• → GrpC• → 0

Show that d1 = 0 if F • is strict. There is in fact a sequence of such mapsd1, d2 . . ., and strictness is equivalent to the vanishing of all of them.

3. When C• = E•(X) with its Hodge filtration, show that Epq1∼= Hq(X,ΩpX)

and that d1 is induced by the α 7→ ∂α on ∂-cohomology. Conclude thatthese maps vanish when X is compact Kahler.

4. Let U3(R) the space of upper triangular 3× 3 matrices1 x z0 1 y0 0 1

with entries in a commutative ring R. The Iwasawa manifold is the quotientU3(C)/U3(Z+Z

√−1). Verify that this is a compact complex manifold with

a nonclosed holomorphic form dz − xdy.

11.3. Hodge decomposition for Moishezon manifolds

A compact complex manifold need not have any nonconstant meromorphicfunctions at all. At the other extreme, a compact manifold X is called Moishezon ifits field of meromorphic functions is large as possible, that is if it has transcendencedegree equal to dimX (this is the maximum possible by a theorem of Siegel [Sh]).This is a very natural class of manifolds which includes smooth proper algebraicvarieties. Moishezon manifolds need not be Kahler, see [Har, appendix B] forexplicit examples, nevertheless theorem 11.2.4 holds for these manifolds. Moregenerally, we have

Theorem 11.3.1. If X is a compact complex manifold for which there existsa compact Kahler manifold and a surjective bimeromorphic map f : X → X, thenthe conclusion of theorem 11.2.4 holds.

11.3. HODGE DECOMPOSITION FOR MOISHEZON MANIFOLDS 145

Corollary 11.3.2. The Hodge decomposition holds for Moishezon manifolds.

Proof. Moishezon [Mo] proved that that there exists a bimeromorphic mapX → X with X smooth projective.

We have already proved a special case of Serre duality for Kahler manifolds. Infact, the result holds for a general compact complex manifold X. There is a pairing

Hq(X,ΩpX)⊗Hn−q(X,Ωn−pX )→ C

induced by

(α, β)→∫X

α ∧ β

Theorem 11.3.3. Suppose X is a compact complex manifold. Then

a) (Cartan) dimHq(X,ΩpX) <∞.b) (Serre) The pairing described above is perfect

Proof. Both results can be deduced from a Hodge decomposition theorem forthe ∂-operator, see [GH].

Proof of theorem 11.3.1. Proofs can be found in [D1, DGMS]. We willbe content to outline a proof. Let n = dimX. There is a map

f∗ : Hq(X,ΩpX)→ Hq(X,ΩpX

)

which is induced by the map α 7→ f∗α of (p, q)-forms. We claim that the map f∗

is injective. To see this, define a map

f∗ : Hq(X,ΩpX

)→: Hq(X,ΩpX),

analogous to the Gysin map, as the dual to the pullback map

Hn−q(X,Ωn−pX )→ Hn−q(X,Ωn−pX

)

We have f∗f∗(α) = α which proves injectivity of f∗ as claimed. By similar reason-ing, f∗Hi(X,C)→ Hi(X,C) is also injective.

We claim that F is strict. As we saw in the previous exercises this is equivalentto the vanishing of the differentials d1, d2 . . .. Since we have only really defined d1,we check only this case, although the same argument works in general. Considerthe commutative diagram

Hq(ΩpX)d1 //

f∗

Hq(Ωp+1X )

f∗

Hq(Ωp

X) d1 // Hq(Ωp+1

X)

Since the bottom d1 vanishes, the same goes for the top.The filtration F can also be shown to be strict. We can now argue as in the

proof of theorem 11.2.4 that the filtrations give a Hodge structure on Hi(X)

Exercises

1. Prove the identity f∗f∗α = α used above.


2. Check the version Serre duality given in theorem 11.3.3 holds for Kahlermanifolds using what we have already proved.

11.4. Hypercohomology*

At this point, it is convenient to give a generalization of the constructions fromchapter 4. Recall that a complex of sheaves is a possibly infinite sequence of sheaves

. . .F i di

−→ F i+1 di+1

−→ . . .

satisfying di+1di = 0. We say that the complex is bounded (below) if and onlyfinitely many of these sheaves are nonzero (or if F i = 0 for i 0). Given anysheaf F and natural number n, we get a bounded complex F [n] consisting of Fin the −nth position, and zeros elsewhere. The collection of bounded (respectivelybounded below) complexes of sheaves on a space X form a category Cb(X) (re-spectively C+(X)), where a morphism of complexes f : E• → F• is defined to bea collection of sheaf maps E i → F i which commute with the differentials. Thiscategory is abelian. We define additive functors Hi : Cb(X)→ Ab(X) by

Hi(F•) = ker(di)/ im(di−1)

A morphism in C+(X) is a called a quasi-isomorphism if it induces isomorphismson all the sheaves Hi.

Theorem 11.4.1. Let X be a topological space, then there are additive functorsHi : C+(X)→ Ab, with i ∈ N, such that

(1) For any sheaf F , Hi(X,F [n]) = Hi+n(X,F)(2) If 0→ E• → F• → G• → 0 is exact, then there is an exact sequence

0→ H0(X, E•)→ H0(X,F•)→ H0(X,G•)→ H1(X, E•)→ . . .

(3) If E• → F• is a quasi-isomorphism, then the induced map Hi(X, E•) →Hi(X,F•) is an isomorphism.

Hi(X, E•) is called the ith hypercohomology group of E•.

Proof. We outline the the proof. Further details can be found in [GM], [I]or [Wl]. We start by redoing the construction of cohomology for a single sheafF . The functor G defined in 4.1, gives a flasque sheaf G(F) with an injective mapF → G(F). C1(F) is the cokernel of this map. Applying G again yields a sequence

F → G(F)→ G(C1(F))

By continuing as above, we get a resolution by flasque sheaves

F → G0(F)→ G1(F)→ . . .

Theorem 5.1.4 shows that Hi(X,F) is the cohomology of the complex Γ(X,G•(F)),and this gives the clue how to generalize the construction. The complex G• isfunctorial. So given a complex

. . .F i d−→ F i+1 . . .

11.4. HYPERCOHOMOLOGY* 147

we get a commutative diagram

. . . F id //

F i+1

. . .

G0(F i) d //

∂

G0(F i+1)

∂

. . . . . .

We define the total complex

T i(F) =⊕p+q=i

Gp(Fq)

with a differential δ = d+ (−1)q∂. We can now define

Hi(X,F•) = Hi(Γ(X, T •(G)))

When applied to F [n], this yields Hi(Γ(X,G•(F))[n]) which as we have seen isHi(X,F), and this proves (1).

(2) can be deduced from the exact sequence

0→ T •(F•)→ T •(G•)→ T •(F•)→ 0

given in the exercises.We now turn to the last statement, and prove it for bounded complexes. For

any complex E• of sheaves (or elements of an abelian category), we can introducethe truncation operator given by the subcomplex

τ≤pE i =

Ep if i < p

ker(Ep → Ep+1) if i = p

0 otherwise

Truncation yields an increasing filtration τ≤p or a decreasing filtration τ≤−p. Thekey property is:

Lemma 11.4.2. There is an exact sequence of complexes

0→ τ≤q−1E• → τ≤qE• → Hq(E•)[−q]→ 0

for each q.

A quasi-isomorphism E• → F• induces a quasi-isomorphism τ≤qE• → τ≤qF•for each q. Thus applying we get a diagram with exact rows

. . . // Hi(τ≤q−1E•) //

Hi(τ≤qE•) //

Hi−q(Hq(E•)) //

∼=

. . .

. . . // Hi(τ≤q−1F•) // Hi(τ≤qF•) // Hi−q(Hq(E•)) // . . .

Thus (3) follows by induction on q.


The precise relationship between the various (hyper)cohomology groups is usu-ally expressed by the spectral sequence

Epq1 = Hq(X, Ep)⇒ Hp+q(E•)There are a number of standard consequences that we can prove directly. The firstis a refinement of theorem 5.1.4.

Corollary 11.4.3. If E• is a bounded complex of acyclic sheaves, Hi(X, E•) =Hi(Γ(X, E•)).

Proof. We sketch the proof, leaving the details for the exercises. The “stupid”filtration

σpE• = E≥p = . . . 0→ Ep → Ep+1 → . . .

is gotten by dropping the first p − 1 terms of the complex. We have an exactsequence

(11.4.1) 0→ E≥p+1 → E≥p → Ep[−p]→ 0

Leading to an exact sequence

Hp(E≥p+1)→ Hp(E≥p)→ H0(Ep)→ Hp+1(E≥p+1)

By descending induction we can show that Hp(E≥p+1) = 0 for all p. Therefore

Hp(E≥p)→ H0(Ep)

Hp+1(E≥p+1)→ H0(Ep+1)are injective. Thus

Hp(E≥p) = ker[H0(Ep)→ H0(Ep+1)]

is exact. Further analysis using (11.4.1) and the acyclity assumption shows that

Hp(X, E•) = . . . = Hp(E≥p−1) =ker[H0(Ep)→ H0(Ep+1)]im[H0(Ep−1)→ H0(Ep)]

Corollary 11.4.4. Suppose that E• is a bounded complex of sheaves of vectorspaces then

dim Hi(E•) ≤∑p+q=i

dimHq(X, Ep)

Proof. The corollary follows by induction on the length (number of nonzeroentries) of E• using the long exact sequences on hypercohomology coming from(11.4.1).

Corollary 11.4.5. Suppose that E• is a bounded complex with Hq(X, Ep) = 0for all p+ q = i, then Hi(E•) = 0.

We can extract on more corollary, using the following elementary result.

Lemma 11.4.6. If

. . . Ai → Bi → Ci → Ai+1 → . . .

is a finite sequence of finite dimensional vector spaces,∑(−1)i dimBi =

∑(−1)i dimAi +

∑(−1)i dimCi

11.5. HOLOMORPHIC DE RHAM COMPLEX* 149

Corollary 11.4.7. If∑

dimHq(X, Ep) ≤ ∞, then∑(−1)i dim Hi(E•) =

∑(−1)p+q dimHq(X, Ep)

In order to facilitate the computation of hypercohomology, we need a criterionfor when two complexes are quasi-isomorphic. We will say that a filtration

E• ⊇ F pE• ⊇ F p+1E• . . .

is finite (of length ≤ n) if E• = F aE• and F a+nE• = 0 for some a.

Lemma 11.4.8. Let f : E• → F• be a morphism of bounded complexes. Supposethat F pE• and GpF• are finite filtrations by subcomplexes such that f(F pE•) ⊆GpF•. If the induced maps

GrpF (E•)→ GrpG(F•)

are quasi-isomorphisms for all p, then f is a quasi-isomorphism.

Exercises

1. If F• is a bounded complex with zero differentials, show that Hi(X,F•) =⊕jHi−j(X,F j).

2. Finish the proof of lemma 11.4.3.3. Prove lemma 11.4.8 by induction on the length.

11.5. Holomorphic de Rham complex*

Let X be a C∞ manifold. We can resolve CX by the complex of C∞ forms E•X .In other words, CX and E•X are quasi-isomorphic. Since E•X is acyclic, it followsthat

Hi(X,CX) = Hi(X,CX [0]) ∼= Hi(X, E•X) ∼= Hi(Γ(X, E•X))

We have just reproved de Rham’s theorem.Now suppose that X is a (not necessarily compact) complex manifold. Then

we define a subcomplex

F pE•X =∑p′≥p

Ep′qX

The image of the mapHi(X,F pE•X)→ Hi(X, E•X)

is the filtration introduced just before theorem 11.2.4. We want to reinterpret thispurely in terms of holomorphic forms. We define the holomorphic de Rham complexby

OX → Ω1X → Ω2

X . . .

We have a natural map Ω•X → E•X which takes σp to F p, where σpΩ•X = Ω≥pX .Dolbeault’s theorem 9.2.3 implies that F p/F p+1 is quasi-isomorphic to σp/σp+1 =ΩpX [−p]. Therefore, lemma 11.4.8 implies that Ω•X → E•X , and more generallyσpΩ•X → F pE•X , are quasi-isomorphisms. Thus:

Lemma 11.5.1. Hi(X,C) ∼= Hi(X,Ω•X) and F pHi(X,C) is the image of Hi(X,Ω≥pX ).


When X is compact Kahler, theorem 11.2.4 implies that the map

Hi(X,Ω≥pX )→ Hi(X,Ω•X)

is injective.From corollaries 11.4.4 , 11.4.5, 11.4.7 we obtain

Corollary 11.5.2. If X is compact, the ith Betti number

bi(X) ≤∑p+q=i

dimHq(X,ΩpX)

and the Euler characteristic

e(X) =∑

(−1)ibi(X) =∑

(−1)p+q dimHq(X,ΩpX)

Corollary 11.5.3. If Hq(X,ΩpX) = 0 for all p+ q = i, then Hi(X,C) = 0.

The next corollary uses the notion of Stein manifold that will be discussed laterin section 15.2. For the time being, we note that Stein manifolds include smoothaffine varieties.

Corollary 11.5.4. Let X be a Stein manifold and particular a smooth affinevariety, then Hi(X,C) = 0 for i > dimX.

Proof. This follows from theorem 15.2.9.

Exercises

1. Suppose that Hi(X,F) = 0 for i > N and any locally free sheaf F . Showthat bi(X) = 0 for i > N + dimX.

2. Show that the inequality in corollary 11.5.2 is strict for the Iwasawa man-ifold (exercises §11.2).

11.6. The Deligne-Hodge decomposition*

We fix the following: a smooth hypersurface (also called a smooth divisor)X ⊂ Y in a projective smooth variety and U = Y −X. Our goal is to understand thecohomology and Hodge theory of U . This can be calculated using C∞ differentialforms E•U , but it will more useful to compute this with forms having controlledsingularities. We define ΩpY (∗X) to be the sheaf of meromorphic p-forms which areholomorphic on U . This is not coherent, but it is a union of coherent subsheavesΩpY (mX) of mermorphic p-forms with at worst poles of order at m along X. Wealso define ΩpY (logX) ⊂ ΩpY (1X) as the subsheaf of meromorphic forms α such thatboth α and dα have simple poles along X. If we choose local coordinates z1, . . . znso that X is defined by z1 = 0. Then the sections of ΩpY (logX), are locally spannedas an OX module by

dzi1 ∧ . . . dzip | ij > 1 ∪dz1 ∧ dzi2 ∧ . . . dzip

z1

Ω•X(logD) ⊂ Ω•Y (∗X) is a subcomplex.

Proposition 11.6.1. There are isomorphisms

Hi(U,C) ∼= Hi(Y,Ω•Y (logX)) ∼= Hi(Y,Ω•Y (∗X))

11.6. THE DELIGNE-HODGE DECOMPOSITION* 151

Proof. Details can be found in [GH, pp 449-454]. The key point is to showthat the inclusions

(11.6.1) Ω•Y (logX) ⊂ Ω•Y (∗X) ⊂ j∗E•Uare quasi-isomorphisms, where j : U → Y is the inclusion. This can be reduced acalculation where Y is replaced by disk with coordinate z, and withX correspondingto the origin. Then (11.6.1) becomes

OD //

d

∑n∈ZODzn //

d

C∞D−0

d

OD dz

z//∑n∈ZODzndz // E1

D−0

d

E2D−0

The cohomology in each columns is C in degrees 0, 1, and horizontal maps induceisomorphisms between these.

The spaces in the proposition on the right carry natural filtrations. The Hodgefiltration:

F pHi+1(U) = im[Hi+1(0→ ΩpY (logX)→ Ωp+1Y (logX) . . .)→ Hi+1(Ω•Y (logX))]

and the pole filtration induced by

PolepHi+1(U) = im[Hi+1(. . . 0→ ΩpY (X)→ Ωp+1Y (2X) . . .)→ Hi+1(Ω•Y (∗X))]

It follows more or less immediately that F p ⊆ Polep. Equality need not hold ingeneral, but it does an important case studied later in section 16.5.

In order to relate this to the cohomology of X, we use residues. We have a map

Res : ΩpY (logX)→ Ωp−1X

call the Poincare residue map given by

Res(α ∧ dz1

z1) = α|X

Res commutes with d, therefore it gives a map of complexes

Ω•Y (logX)→ Ω•X [−1]

where [−1] indicates a shift of indices by −1. This induces a map

Hi(U,C) = Hi(Ω•Y (logX))→ Hi−1(X,C)

After normalizing this by a factor of 12π√−1

, it takes integer cohomology to integercohomology (modulo torsion). This can be described topologically as a composition

Hi(U)→ Hi(Tube) ∼→ Hi−1(X)

where Tube is a tubular neighbourhood and the second map is the inverse of theThom isomorphism §5.4. The residue map is an epimorphism of sheaves, and thekernel is precisely the sheaf of holomorphic forms. So we have an exact sequence

(11.6.2) 0→ Ω•Y → Ω•Y (logX)→ Ω•X [−1]→ 0


which leads to a long exact sequences

(11.6.3) . . . Hq(ΩpY )→ Hq(ΩpY (logX))→ Hq(Ωp−1X )→ Hq+1(ΩpY ) . . .

(11.6.4) . . . Hi(Y,C)→ Hi(U,C)→ Hi−1(X,C)→ Hi+1(Y,C) . . .

The second is called the Gysin sequence.

Theorem 11.6.2 (Deligne). The Hodge filtration on on Hi(U,C) is strict, i.ethe maps

Hi+1(0→ ΩpY (logX)→ Ωp+1Y (logX) . . .)→ Hi+1(Ω•Y (logX))

are injective. In particular, there is a (noncanonical) decomposition

Hi(U,C) ∼=⊕p+q=i

Hq(Y,ΩpY (logX))

Proof. By corollary 11.2.3, it is enough to prove that

dimHi(U) =∑p+q=i

dimHq(Y,ΩpY (logX))

Form (11.6.3) and (11.6.4), we get

dimHq(ΩY (logX)) = dim ker[Hq(Ωp−1X )→ Hq+1(ΩpY )]+dim im[Hq−1(Ωp−1

X )→ Hq(ΩpY )]

and

dimHi(U) = dim ker[Hi−1(X)→ Hi+1(Y )] + dim im[Hi−2(X)→ Hi(Y )]

Combining the last equation with corollaries 11.2.5 and 11.2.10. shows that

dimHi(U) =∑

dim ker[Hq(Ωp−1X )→ Hq+1(ΩpY )] +

∑dim im[Hq−1(Ωp−1

X )→ Hq(ΩpY )]

=∑

dimHq(Y,ΩpY (logX))

Corollary 11.6.3 (Kodaira Vanishing). If Y ⊂ PN an n-dimensional non-singular variety, then Hi(Y,ΩnY ⊗OY (1)) = 0 for i > 0.

Proof. By Bertini’s theorem, we can choose a hyperplane H ⊂ Pn such thatX = Y ∩H is smooth. Then by the theorem and corollary 11.5.4

dimHi(Y,ΩnY (logX)) ≤ dimHi+n(U,C) = 0

when i > 0. A direct calculation shows that

ΩnY (logX) = ΩnY (X) ∼= ΩnY ⊗OY (1)

Remark 11.6.4. Deligne [D2] showed strictness when X is a divisor with nor-mal crossings. This is a much harder theorem which was proved on route to con-structing a mixed Hodge structure on Hi(U). This would take another book toexplain. Fortunately, Peters and Steenbrink [PS] have written one.

Remark 11.6.5. Kodaira proved a stronger statement, where O(1) is replacedby an ample line bundle, by an entirely different method (cf [GH]). However thestronger form can actually be reduced to the above statement.

Exercises

11.6. THE DELIGNE-HODGE DECOMPOSITION* 153

1. Work out (11.6.3) and (11.6.4) explicitly when Y = P2 and X is a smoothcurve of degree d.

2. Show that forms in H0(Y,ΩpY (logX)) are closed. Is this necessarily truefor forms in H0(Y,ΩpY (kX))?

3. Verify the isomorphism ΩnY (logX) ∼= ΩnY ⊗ OY (1) used in the proof ofcorollary 11.6.3.

CHAPTER 12

A little algebraic surface theory

Now we want to return to geometry, armed with what we have learned in theprevious chapters. We have already looked at Riemann surfaces (which will bereferred to as complex curves from now on) in some detail. So we consider the nextstep up. A nonsingular complex surface is a two dimensional complex manifold. Byan algebraic surface, we will mean a two dimensional nonsingular projective variety.In this chapter, we will present a somewhat unsystematic account of surface theory.

12.1. Examples

The basic discrete invariant of a curve is its genus. For algebraic surfaces,there are several numbers which play a similar role. We can use the Hodge num-bers. By the symmetry properties considered earlier, there are only 3 that matterh10, h20, h11. The first two are traditionally called (and denoted by) the irregularity(q = h10) and geometric genus (pg = h20). The basic topological invariants are theBetti numbers and that Euler characteristic, and these can expressed in terms ofthe previous invariants

b1 = b3 = 2q

b2 = 2pg + h11

and

e = 2− 2q + 2pg + h11

In some cases, it may be more convenient to calculate other subsets of these num-bers, and solve.

Example 12.1.1. The basic example is X = P2. We have q = pg = 0 andh11 = 1. In fact, as Hodge structures H1(P2) = 0 and H2(P2) = Z(−1).

Example 12.1.2. The next example is a ruled surface over a smooth projectivecurve C of genus g. This is the P1-bundle P(V ) associated to a rank 2 vector bundleV over C. When C = P1, these are all of the form Fn = P(O ⊕O(n)), n ≥ 0. Fncan also be described as the toric variety (section 3.5) associated to the fan

155

156 12. A LITTLE ALGEBRAIC SURFACE THEORY

(1,0)

(0,1)

σ1

2σ

σ3

σ

(−1, n)

4

The invariants are q = g, pg = 0 and h11 = 2.

Example 12.1.3. If X = C1 × C2 is a product of two nonsingular curves ofgenus g1 and g2. Then by Kunneth’s formula,

H1(X) ∼= H1(C1)⊕H1(C2)

H2(X) ∼= [H2(C1)⊗H0(C2)]⊕ [H1(C1)⊗H1(C2)]⊕ [H0(C1)⊗H2(C2)]as Hodge structures. Therefore q = g1 + g2, pg = g1g2 and h11 = 2g1g2 + 2.

Example 12.1.4. Let X ⊂ P3 be a smooth surface of degree d. Then q = 0.We will list the first few values of the remaining invariants.

d pg h11

2 0 23 0 74 1 205 4 456 10 86

These can be calculated using formulas given later (16.3.4).

A method of generating new examples from old is by blowing up. We start bydescribing the blow up of C2 at 0:

Bl0C2 = (x, `) ∈ C2 × P1 |x ∈ `The projection p1 : Bl0C2 → C2 is one to one away from 0 ∈ C2. This can begeneralized to yield the blow up BlpX → X of a surface X at point p. Let B ⊂ Xbe a coordinate ball centered at p. After identifying B with a ball in C2 centeredat 0, we can form let Bl0B be the preimage of B in Bl0C2. The boundary of Bl0Bcan be identified with the boundary of B. Thus we can glue X − B ∪ Bl0B toform BlpX. When X is algebraic, BLpX is again algebraic. Although this is notobvious from the above description.

Let us compute H∗(BlpX,Z). Set Y = BlpX and compare Mayer-Vietorissequences

Hi(X) //

Hi(X −B′)⊕Hi(B) //

Hi(X −B′ ∩B)

=

Hi(Y ) // Hi(Y −BlpB′)⊕Hi(BlpB) // Hi(Y −BlpB′ ∩BlpB)

where B′ ⊂ B is a smaller ball. It follows that

12.1. EXAMPLES 157

Lemma 12.1.5. H1(BlpX) ∼= H1(X) and H2(BlpX) = H2(X)⊕ Z.

Corollary 12.1.6. q and pg are invariant under blowing up. h11(BlpX) =h11(X) + 1.

Proof. The lemma implies that b1 = 2q is invariant and b2(Y ) = b1(X) + 1.Since b2 = 2pg + h11, the only possibilities are h11(Y ) = h11(X) + 1, pg(Y ) =pg(X), or that pg(Y ) < pg(X). The last inequality means that there is a nonzeroholomorphic 2-form on X that vanishes on X − p, but this is impossible.

A morphism of varieties f : V →W is called birational if it is an isomorphismwhen restricted to nonempty Zariski open sets of X and Y . A birational mapV 99K W is simply an isomorphism of Zariski open sets. Blow ups and theirinverses (“blow downs”) are examples of birational morphisms and maps. For therecord, we point out

Theorem 12.1.7 (Castelnuovo). Any birational map between algebraic surfacesis a given by a finite sequence of blow ups and downs.

Proof. See [Har, BPV].

Therefore, we get the birational invariance of q and pg. However, there areeasier ways to prove this. Blowing up of singular points figures in the proof of thenext important theorem.

Theorem 12.1.8 (Zariski). Given singular algebraic surface Y , there is a non-singular surface X and a morphism π : X → Y , called a resolution of singularities,which is an isomorphism away from the singular points.

Proof. [BPV].

Corollary 12.1.9 (Zariski). If f : X 99K V rational function from an alge-braic surface. Then there is a finite sequence of blow ups such that Y → X suchthat f extends to a holomorphic map f ′ : Y → V .

Proof. We can construct Y by resolving singularities of the closure of thegraph of f .

An elliptic surface is a surface X which admits a surjective morphism f : X →C to a smooth projective curve such that all nonsingular fibres are elliptic curves.

Example 12.1.10. A simple example of an elliptic surface is given as fol-lows: choose two distinct nonsingular cubics E0, E1 ⊂ P2 defined by f0(x, y, z)and f1(x, y, z). These generate a pencil of cubics Et = V (tf1 + (1− t)f0). Define

X = (p, t) ∈ P2 × P1 | p ∈ EtProjection to P1 makes this an elliptic surface. X can be identified with the blowup of P2 at the nine points E0 ∩ E1. So q = pg = 0 and h11 = 10.

Example 12.1.11. Consider the family of elliptic curves in Legendre form

E = ([x, y, z], t) ∈ P2 × C− 0, 1 | y2z − x(x− z)(x− tz) = 0 → CThe above equation is meaningful if t = 0, 1, and it defines a rational curve with asingle node. By introducing s = t−1, we get an equation

sy2z − x(x− z)(sx− z) = 0


which defines a union of lines when s = 0. In this way, we can extend E toa surface E ′ → P1. Unfortunately, E ′ is singular, and it is necessary to resolvesingularities to get an nonsingular surface E containing E (we can take the minimaldesingularization, which for our purposes means that b2(E) is chosen as small aspossible).

Exercises

1. Compare the construction of the blow up given here and in section 3.5(identify ((x, t) with ((x, xt), [t, 1])).

2. Finish the proof of lemma 12.1.5.3. Show that there is a nonsingular quartic X ⊂ P3 containing a line `, which

we can assume to be x2 = x3 = 0. Show that the map P3 99K P1 definedby [x0, . . . x3] 7→ [x0, x1] determines a morphism X → P1 which makes itan ellptic surface.

4. Given a smooth projective curve C, the symmetric product S2C = C×C/σwhere σ is the involution interchanging factors. This has the structure ofsmooth algebraic surface such that Hi(S2C,Q) = Hi(C×C,Q)σ. Computethe Betti and Hodge numbers for S2C.

12.2. The Neron-Severi group

Let X be an algebraic surface. The image of the first Chern class map is theNeron-Severi group NS(X) = H2(X,Z)∩H11(X). The rank of this group is calledPicard number ρ(X). We have ρ ≤ h11 with equality if pg = 0.

A divisor on X is a finite integer linear combination∑niDi of possibly singular

irreducible curves Di ⊂ X. We can define a line bundle OX(D) as we did forRiemann surfaces in section 6.5. If fi are local equations of Di ∩ U in some openset U ,

OX(D)(U) = OX(U)1

fn11 fn2

2 . . .

is a fractional ideal. In particular, if D is irreducible OX(−D) is the ideal sheaf ofD.

Lemma 12.2.1. If Di are smooth curves, then c1(OX(∑niDi)) =

∑ni[Di].

Proof. This is an immediate consequence of theorem 7.5.7.

WhenD is singular, we can simply define its fundamental class to be c1(OX(D)).The cup product pairing

H2(X,Z)×H2(X,Z)→ H4(X,Z)→ Z

restricts to a pairing on NS(X) denoted by “·”.

Lemma 12.2.2. Given a pair of transverse smooth curves D and E,

D · E =∫X

c1(O(D)) ∪ c1(O(E)) =∫D

c1(OX(E))|D = #(D ∩ E)

Proof. This follows from the lemma 12.2.1 and proposition 5.5.3. The numberip(D,E) is seen to be always +1 in this case.

12.2. THE NERON-SEVERI GROUP 159

If the intersection of the curves D and E is finite but not transverse, it is stillpossible to give a geometric meaning to the above product. Choose local coordinatescentered at p, and local equations f and g for D and E respectively.

Definition 12.2.3. The local intersection number, ip(D,E) = dimOp/(f, g).(This only depends on the ideals (f) and (g), so it is well defined.)

Proposition 12.2.4. If D,E are curves such D ∩ E is finite, then

D · E =∑p∈X

ip(D,E)

Proof. We give the proof when D and E are smooth. As in the proof ofproposition 5.5.3, the number on the left is given by∫

X

τD ∧ τE

for appropriate representative τD, τE for the Thom classes. In particular, we assumethe supports are small enough that it breaks up into a sum of integrals over disjointcoordinate neighbourhoods of p ∈ D ∩ E.

We need a convenient expression for the Thom classes. We first note that if ρ :R+ → [0, 1] is cutoff function: 0 in a neighbourhood of 0 and 1 in a neighbourhoodof∞, then 1

2πdρ(r)∧dθ gives a local expression for the Thom class of 0 ∈ R2. Thusafter choosing local equations of D,E at p ∈ D ∩ E as above, we can assume that(locally) τD = τf and τE = τg, where

τf =1

2π√−1

dρ(|f |) ∧ dff

Let h(z1, z2) = (f(z1, z2), g(z1, z2)). It maps a small ball 0 ∈ U ⊂ C2 to anothersmall ball 0 ∈ U ′. The degree of h is the number of points in the fibre h−1(y) foralmost all y. This coincides with ip(E,D) by lemma 1.3.2. Computing the integralby a change of variables gives∫

U

τf ∧ τg =∫U ′h∗(τf ∧ τg)

= (deg h)∫τz1 ∧ τz2

= deg h = ip(E,D)

Recall that H2(P2,Z) = H4(P2,Z) = Z, and the generator of H2(P2,Z) is theclass of line [L]. Given a curve D defined by a polynomial f , we define degD =deg f . Then [D] = degD[L].

Corollary 12.2.5 (Bezout). If D,E are curves on P2 with a finite intersec-tion, ∑

p∈Xip(D,E) = deg(D) deg(E)

Proof. We have [D] = degD[L], and [E] = degE[L], so that D · E =degD degE(L · L) = degD degE.

Corollary 12.2.6. If D,E are distinct irreducible curves, then D ·E ≥ 0, andequality holds only if they are disjoint.


This corollary can fail when D = E. For example by corollary 5.6.3, thediagonal in a product of curves ∆ ⊂ C × C has negative self-intersection as soonas the genus of C is greater than 1. From lemma 12.2.1, we obtain:

Lemma 12.2.7. If D is a smooth curve

D2 = D ·D =∫D

c1(OX(D)) = deg(OD(D))

Given a surjective morphism f : X → Y of algebraic surfaces. The pullbackf∗ : H2(Y,Z) → H2(X,Z) preserves the Neron-Severi group and the intersectionpairing. This can be interpreted directly in the language of divisors. Given anirreducible divisor D on Y . We can make the set theoretic preimage f−1D intoa divisor by pulling back the ideal, i.e. OX(−f−1D) = OY (−D). We extendthis operation to all divisors by linearity. The operation satisfies f∗OY (D) =OX(f−1D). Since c1 is functorial, we get

Lemma 12.2.8. f−1 is compatible with f∗ on NS(X).

Exercises

1. Let X = C × C be product of a curve with itself. Consider, the divisorsH = C×p, V = p×C and the diagonal ∆. Compute their intersectionnumbers, and show that these are linearly independent in NS(X)⊗Q. Thusthe Picard number is at least 3. Show that this is at least 4 if C admits anontrivial automorphism with the appropriate conditions.

2. E = C/(Z + Zτ) be an elliptic curve, and let X = E × E. Show that thePicard number is 3 for “most” τ , but that it is 4 for τ =

√−1.

3. Let π : Y → P2 be the blow up of P2 at point p. Let L1 be a line in P2

containing p and 2 another line not containing p. Show that π∗L1 = π∗L2

and π∗L1 = E+F where E = π−1(p) and F is the closure in Y of L1−p(F is called the strict transform of L1). Use all of this to show that E2 =−1.

4. A divisor is called (very) ample if OX(H) is. If H is ample, then provethat H2 > 0 and that H ·C > 0 any curve C ⊂ X. This pair of conditionscharacterizes ampelness (Nakai-Moishezon). Show that the first conditionalone is not sufficient.

12.3. Adjunction and Riemann-Roch

In this section, we introduce two of the most basic tools of surface theory. Thefirst result, called the adjunction formula, gives the genus of curve on a surface. Toset things up, recall that the canonical divisor K of a smooth projective curve is adivisor is such that OC(K) ∼= Ω1

C . Since this determines K uniquely up to linearequivalence, so we can talk about the canonical divisor class KC . A canonicaldivisor KX (class) on a surface X is divisor such that OX(KX) ∼= Ω2

X . The linearequivalence class is again well defined. For the present, we only need its image inNS(X), and this can be defined to be c1(Ω2

X).

Theorem 12.3.1. If C is a smooth curve of genus g on an algebraic surfaceX, then

2g − 2 = (KX + C) · C

12.3. ADJUNCTION AND RIEMANN-ROCH 161

Proof. From (11.6.2), we obtain an exact sequence

(12.3.1) 0→ Ω2X → Ω2

X(logC)→ Ω1C → 0

where Ω2X(logC) is spanned locally by dz1∧dz2

f , where f is local equation for C. Itfollows that we can identify this with Ω2

X ⊗ OX(C). The holomorphic forms Ω2X

are spanned locally f dz1∧dz2f . Thus we can identify the inclusion in (12.3.1) withthe tensor product of the map OX(−C)→ OX with Ω2

X ⊗OX(C). The cokernel ofthe map just described is OC ⊗Ω2

X , that is the restriction of Ω2X ⊗OX(C) to C. So

in summary, Ω1C is isomorphic to the restriction of Ω2

X ⊗OX(C) to C. Therefore∫C

c1(Ω1C) =

∫C

c1(Ω2X ⊗OX(C))

=∫C

(c1(Ω2X) + c1(OX(C))

= (KX + C) · CThe left side is the degree of the KC , but this is 2g − 2 by proposition 6.5.7.

We can use this to recover the formula for the genus of a degree d curve C ⊂ P2.Since NS(P2) ⊆ H2(P2) = Z, we can identify KP2 with an integer k. Therefore

g =12

(k + d)d+ 1

When d = 1, we know the g = 0 so that k = −3. Thus g = (d− 1)(d− 2)/2.A fundamental, and rather difficult, problem in algebraic geometry is to esti-

mate dimH0(X,OX(D)). As a first step, one can calculate the Euler characteristic

χ(OX(D)) =∑

(−1) dimHi(X,OX(D))

by the Riemann-Roch formula given below. The higher cohomologies can then becontrolled in some cases by other techniques. The advantage of the Euler charac-teristic is the additivity property:

Lemma 12.3.2. If 0 → F1 → F2 → F3 → 0 is an exact sequence of sheaveswith

∑dimHi(X,Fj) <∞, then χ(F2) = χ(F1) + χ(F3).

Proof. This follows from lemma 11.4.6.

Theorem 12.3.3 (Riemann-Roch). If D is a divisor on a surface X,

χ(OX(D)) =12D · (D −KX) + χ(OX)

Proof. We prove this under the assumption that D =∑i niDi is a sum of

smooth curves. (In fact, by a simple trick, it is always possible to reduce to thiscase. The basic idea can be found, for example, in the proof of [Har, chap V, thm1.1].) By induction, it suffices to prove Riemann-Roch for D = D′ ± C, where theformula holds for D′ and C is smooth. The idea is to do induction, so we will usethe Riemann-Roch theorem for C as given in the exercises of §6.5. We treat thecase of D = D′ + C, leaving the remaining case for the exercises. Tensoring thesequence

0→ OX(−C)→ OX → OC → 0by O(D) yields

0→ OX(D′)→ OX(D)→ OC(D)→ 0


Therefore using this together with the adjunction formula and Riemann-Roch onC

χ(OX(D)) = χ(OX(D′)) + χ(OC(D))

=12D′(D′ −K) + χ(OX) + deg(D|C) + 1− g(C)

=12D′(D′ −K) + C ·D − 1

2C(C +K) + χ(OX)

=12D(D −K) + χ(OX)

Exercises1. Find a formula for the genus of a curve in P1 × P1 in terms of its bidegree.2. Given a morphism f : C → D, calculate the self-intersection of the graph

Γ2f .

3. Do the case D = D′ − C in the proof of Riemann-Roch.4. Find a formula for D ·C in terms of χ(O(D)), χ(O(C)), χ(O(C +D)) andχ(OX).

5. Use the formula of the previous exercise to give another proof of proposition12.2.4.

12.4. The Hodge index theorem

Let X a compact Kahler surface. Then the Kahler form ω is a closed real (1, 1)form. Therefore the Kahler class [ω] is an element of H11(X) ∩H2(X,R).

Theorem 12.4.1. Let X be a compact Kahler surface. Then the restriction ofthe cup product to (H11(X) ∩H2(X,R)) ∩ (R[ω])⊥ is negative definite.

Proof. Locally, we can find an orthonormal basis φ1, φ2 of E(1,0). In thisbasis,

ω =√−12

(φ1 ∧ φ1 + φ2 ∧ φ2)

and the volume form

dvol =ω2

2=(√−12

)2

φ1 ∧ φ1 ∧ φ2 ∧ φ2

using the exercises from 10.2. Choose an element α ∈ (H11(X) ∩ H2(X,R)) andrepresent it by a harmonic real (1, 1) form, then

α =∑

aijφi ∧ φj

with aji = aij . If α is also chosen in (R[ω])⊥, then

α ∧ ω =2√−1

(a11 + a22)dvol

is exact. This is impossible unless a11 + a22 = 0. Therefore∫X

α ∧ α = 2(

2√−1

)2 ∫X

(|a11|2 + |a12|2)dvol < 0

12.5. FIBRED SURFACES 163

Corollary 12.4.2. If H is an ample divisor on an algebraic surface, the in-tersection pairing is negative definite on (NS(X)⊗ R) ∩ (R[H])⊥

Proof. By corollary 10.1.8, [H] is a Kahler class.

Therefore when the intersection form is diagonalized, the diagonal consists ofexactly one +1 and the remaining entries are −1.

Corollary 12.4.3. If H,D are divisors on an algebraic surface such thatH2 > 0 and D ·H = 0, then D2 < 0 unless [D] = 0.

Exercises

1. Prove that the restriction of the cup product to (H20(X) + H02(X)) ∩H2(X,R) is positive definite.

2. Conclude that (the matrix representing) the cup product pairing has 2pg+1positive eigenvalues. Therefore pg is a topological invariant.

3. Let f : X → Y be a morphism from a smooth algebraic surface to apossibly singular projective surface. Consider the set Di of irreduciblecurves which map to points under f . Prove a theorem of Mumford thatthe matrix (Di ·Dj) is negative definite.

12.5. Fibred surfaces

Let us say that surface X is fibred if it admits a nonconstant holomorphicmap f : X → C to a nonsingular curve. For example ruled surfaces and ellipticsurfaces are fibred. Not all surfaces are fibred, however any surface can be fibredafter blowing up: Choose a nontrivial rational function X 99K P1, then blow upX, to get a morphism. The fibre over p ∈ C is the closed set f−1(p). Let z be alocal coordinate at p. Then f−1(p) is defined by the equation f∗z = 0. We canregard f−1(p) as a divisor

∑niDi, where Di are its irreducible components and ni

is the order of vanishing of f∗z along Di. Call a divisor vertical if its irreduciblecomponents are contained in the fibres.

Lemma 12.5.1. Suppose that f : X → C is a fibred surface(a) If F = f−1(p), and D is a vertical divisor, then F ·D = 0. In particular,

F 2 = 0.(b) If D is vertical, then D2 ≤ 0.

Proof. Since the cohomology class of f−1(p) is independent of p, we canassume that F and D are disjoint. This proves (a).

Suppose that D2 > 0, when combined with (a), we would get a contradictionto the Hodge index theorem.

Corollary 12.5.2. A necessary condition for a surface to be fibred is thatthere is a effective divisor with F 2 = 0. In particular, P2 is not fibred.

We can give complete characterization of surfaces fibred over curves of genusgreater than one.

Theorem 12.5.3 (Castenuovo-de Franchis). Suppose that X is an algebraicsurface. A necessary and sufficient condition for X to admit a nonconstant holo-morphic map to a smooth curve of genus g ≥ 2 is that there exists two linearindependent forms ωi ∈ H0(X,Ω1

X) such that ω1 ∧ ω2 = 0.


Proof. The necessity is clear. We will sketch the converse, which gives anice application of corollary 11.2.6. A complete proof can be found in [BPV, pp123-125]. Choosing local coordinates, we can write

ωi = fi(z1, z2)dz1 + gi(z1, z2)dz2

The condition ω1 ∧ ω2 = 0 forces

(f1g2 − f2g1)dz1 ∧ dz2 = 0

Therefore f2/f1 = g2/g1, call this F . Thus ω2 = Fω1. Since ωi are globally defined,F = ω2/ω1 defines a global meromorphic function X 99K P1. By corollary 12.1.9,there exists Y → X which is a composition of blow ups such that F extends to aholomorphic function F ′ : Y → P1. The fibres of F ′ need not be connected. Stein’sfactorization theorem [Har] shows that the map can be factored as

YΦ→ C → P1,

where Φ has connected fibres and C → P1 is a finite to one map of smooth projectivecurves. To avoid too much notation, let us denote the pullbacks of ωi to Y by ωias well. We now have a relation ω2 = Φω1.

We claim that ωi are pullbacks of holomorphic 1-forms on C. We will checkthis locally around a general point p ∈ X. Since ωi are closed (corollary 11.2.6),

(12.5.1) dω2 = dΦ ∧ ω1 = 0

Let t1 be a local coordinate centered at Φ(p). Let us also denote the pullback ofthis function to neighbourhood of p by t1. Then we can choose a function t2 suchthat t1, t2 give local coordinates at p. (12.5.1) becomes dt1 ∧ω1 = 0. Consequentlyω1 = f(t1, t2)dt1, for some function f . The relation dω1 = 0, implies f is a functionof t1 alone. Thus ω1 is locally the pullback of a 1-form on C as claimed. The samereasoning applies to ω2. This implies that the genus of C is at least two.

The final step is to prove that blow up was unnecessary. Let

Y = Y1π1−→ Y2 → . . . X

be a composition of blow ups at points pi ∈ Yi. Then π−11 (p2) ∼= P1. Any map from

P1 to C is constant since it has positive genus. So we conclude that Y → C factorsthrough Y1, and then likewise through Y2 and so on until we reach Y .

An obvious corollary is:

Corollary 12.5.4. If q ≥ 2 and pg = 0, then X admits a nonconstant mapto a curve as above.

Exercises

1. Given an elliptic surface X, show that KX is vertical. Conclude thatK2X ≤ 0.

2. Show that X maps onto a curve of genus ≥ 2 if q > pg+1. (This bound canbe sharpened to (pg + 3)/2 but the argument is more delicate, c.f. [BPV,IV, 4.2].)

CHAPTER 13

Topology of families

13.1. Fibre bundles

A C∞ map f : X → Y of manifolds is called a fibre bundle if it is locally aproduct of Y with another manifold F (called the fibre). This means that thereis an open cover Ui and diffeomorphisms f−1Ui ∼= Ui × F compatible with theprojections. If X = Y × F , the bundle is called trivial. Nontrivial bundles overS1 can be constructed as follows. Let F be a manifold with a diffeomorphismφ : F → F , then glue F × 0 in F × [0, 1] to F × 1 by identifying (x, 0) to(φ(x), 1). This includes the familiar example of the Mobius strip where F = R andφ is multiplication by −1. If the induced map φ∗ : H∗(F ) → H∗(F ), called themonodromy transformation, is nontrivial then the fibre bundle is nontrivial.

A C∞ map f : X → Y is called a submersion if the map on tangent spacesis surjective. The fibres of such a map are submanifolds. A continuous map oftopological spaces is called proper if the preimage of any compact set is compact.

Theorem 13.1.1 (Ehresmann). Let f : X → Y be a proper submersion of C∞

connected manifolds. Then f is a C∞ fibre bundle; in particular, the fibres arediffeomorphic.

Proof. A complete proof can be found in [MK, 4.1]. We sketch a proof thatthe fibres are diffeomorphic. Since Y is connected, we can join any two points, say 0and 1 by a path. Thus we may replace Y by R. Choose a Riemannian metric on X.The gradient ∇f is defined as the vector field dual to df under the inner productassociated to the metric. By assumption df and therefore ∇f is nowhere zero. Theexistence and uniqueness theorem of ordinary linear differential equations allowsus to define, for each p ∈ f−1(0), a C∞ path γp : [0, 1] → X passing through p attime 0 and with velocity ∇f . Then the gradient flow p 7→ γp(1) gives the desireddiffeomorphism.

Exercises

1. Show that theorem can fail for nonproper maps

13.2. Monodromy of families of elliptic curves

Let us look some complex analytic examples. Let Γ be the semidirect product

Z2 o Z where a ∈ Z acts by the matrix(

1 a0 1

). More explicitly, the elements are

triples (m,n, a) ∈ Z3 with multiplication

(m,n, a) · (m′, n′, a′) = (m+m′ + an′, n+ n′, a+ a′)

165

166 13. TOPOLOGY OF FAMILIES

Let Γ act on C×H by

(m,n, a) : (z, t) 7→ (z +m+ nτ, τ + a)

The quotient gives a holomorphic family of elliptic curves

E → H/Zτ→q=exp(−2πiτ)∼= D∗

over the punctured disk. As a C∞ map, it is a fibre bundle with a torus T asfibre. However, it is not locally trivial holomorphically since the fibres are notbiholomorphic. The fibre over τ , Eτ is given by the Weierstrass equation

y2 = 4x3 − g2(τ)x− g3(τ)

This equation makes sense in the limit as τ → ∞ (or equivalently as q → 0),and it defines a rational curve with a node. In this way E → D∗ can be completedto a map of complex manifolds E → D. We have Eτ ∼= C/(τ−1Z + Z). Let a(τ) bethe image in Eτ of the line segment joining 0 to 1, and let b(τ) be the image of thesegment joining 0 to τ−1. If we orient these curves so that a · b = 1, they form abasis of H1(Et,Z). The curve b(t) is called a vanishing cycle since it shrinks to thenode as t→ 0, see figure 1

t=0 t general

b(t)

Figure 1. Vanishing cycle

The restriction of E to a bundle over the circle S = t | |t| = ε is a fibre bundlewith monodromy transformation µ given by the Picard-Lefschetz formula:

µ(a) = a+ b, µ(b) = b.

From a more topological point of view, E|S can be described by taking a trivialtorus bundle T × [0, 1] and gluing the ends T × 0 and together T × 1 using aso called Dehn twist about the vanishing cycle b = b(t). This is a diffeomorphism

13.2. MONODROMY OF FAMILIES OF ELLIPTIC CURVES 167

which is the identity outside a neighbourhood U of b and twists “once around”along b (see figures 2 and 3, U is the shaded region).

Figure 2. T foliated by meridians

Figure 3. T foliated by images of meridians under Dehn twist


Next, consider the family of elliptic curves in Legendre form

E = ([x, y, z], t) ∈ P2 × C− 0, 1 | y2z − x(x− z)(x− tz) = 0 → CThis example was considered earlier, where we completed it to projective surface,and then resolved singularities to obtain a nonsingular surface E → P1. Let’s seehow to calculate the monodromy of going around 0 and 1. The fibres over thesepoints are no longer nodal curves, so the Picard-Lefschetz formula will not apply.However, it is possible to calculate this from a different point of view. Recall thatH/Γ(2) = P1 − 0, 1,∞. E can also be realized as a quotient of C × H by anaction of the semidirect product Z2 o Γ(2) as above. The Γ(2) action extends toH∗ = H ∪ Q ∪ ∞. We can choose a fundamental domain for Γ(2) in H∗ asdepicted in figure 4, the three cusps are 0, 1,∞.

cusp

Figure 4. Fundamental domain of Γ(2)

The subgroup of Γ(2) which fixes ∞ is generated by(1 20 1

)and it follows easily that this is the monodromy matrix for it. We will call E theelliptic modular surface of level two.

Exercises

1. Calculate the monodromy matrices for the remaining cusps for Γ(2).

13.3. Local systems

In this section, we give a more formal treatment of monodromy.Let X be a topological space, a path from x ∈ X to y ∈ X is a continuous map

γ : [0, 1] → X such that γ(0) = x and γ(1) = y. Two paths γ, η are homotopic if

13.3. LOCAL SYSTEMS 169

there is a continuous map Γ : [0, 1] × [0, 1] → X such that γ(t) = Γ(t, 0), η(t) =Γ(t, 1) Γ(0, s) = x and Γ(1, s) = y. This is an equivalence relation. We can composepaths: If γ is a path from x to y and η is a path from y to z, then γ · η is the pathgiven by following one by the other at twice the speed. More formally,

γ · η(t) =

γ(2t) if t ≤ 1/2η(2t− 1) if t > 1/2

This operation is compatible with homotopy in the obvious sense, and the inducedoperation on homotopy classes is associative. We can define a category Π(X) whoseobjects are points of X and whose morphisms are homotopy classes of paths. Thismakes Π(X) into a groupoid which means that every morphism is an isomorphism.In other words every (homotopy class of a) path has a inverse. This is not a groupbecause it is not possible to compose any two paths. To get around this, we canconsider loops, i.e. paths which start and end at the same place. Let π1(X,x) bethe set of homotopy classes of loops based (starting and ending ) at x. This is justHomΠ(X)(x, x) and as such it inherits a composition law which makes it a groupcalled the fundamental group of (X,x). We summarize the standard propertiesthat can be found in almost any algebraic topology textbook, e.g. [Hat, Sp]:

1. π1 is a functor on the category of path connected spaces with base pointand base point preserving continuous maps.

2. If X is path connected, π1(X,x) ∼= π1(X, y) (consequently, we usuallysuppress the base point).

3. Two homotopy equivalent path connected spaces have isomorphic funda-mental groups.

4. Van Kampen’s theorem: If X is a path connected simplicial complex whichis the union two subcomplexes X1 ∪X2, then π1(X) is the free product ofπ1(X1) and π1(X2) amalgamated along the image of π1(X1 ∪X2).

5. If X is a connected locally path connected space, then has a universal coverπ : X → X. π1(X) is isomorphic to the group of deck transformations, i.e.self-homeomorphisms of X which commute with π.

This already suffices to calculate the examples of interest to us. It is easy tosee that the fundamental group of the circle R/Z is Z. The complement in C of aset S of k points is homotopic to a wedge of k circles. Therefore π1(C−S) is a freegroup on k generators.

Let X be a topological space. A local system of Abelian groups is a functorF : Π(X) → Ab. A local system F gives rise to a π1(X,x)-module i.e. Abeliangroup F (x) with a π1(X,x) action. We define a sheaf F as follows

F(U) = f : U →⋃x

F (x) | f(x) ∈ F (x) and ∀γ ∈ Π(U), f(γ(1)) = F (γ)(f(γ(0)))

If every point x ∈ X has an neighbourhood Ux with trivial fundamental group (i.e.X is locally simply connected) then the restrictions F|Ux

are constant. We say thatsuch a sheaf F is locally constant.

Theorem 13.3.1. Let X be a path connected locally simply connected topologicalspace. There is an equivalence of categories between

(1) The category of π1(X)-modules.(2) The category of local systems and natural transformations.(3) The full subcategory of Sh(X) of locally constant sheaves on X.


Proof. See [I, IV,9] , [Sp, p. 360] or the exercises.

In view of this theorem, we will treat local systems and locally constant sheavesas the same.

Let f : X → Y be a fibre bundle. We are going to construct a local systemwhich takes y → Hi(Xy,Z). Given a path γ : [0, 1] → Y joining y0 and y1, thepullback γ∗X = (x, t) | f(x) = γ(t) will be a trivial bundle over [0, 1]. Thereforeγ∗X will deformation retract onto both Xy0 and Xy1 , and so we have isomorphisms

Hi(Xy0) ∼← Hi(γ∗X) ∼→ Hi(Xy1)

The map Hi(Xy0) → Hi(Xy1) can be seen to depend only on the homotopy classof the path, thus we have a local system. When y0 = y1, this is the monodromytransformation considered earlier. The corresponding locally constant sheaf whichwill be described in the next section.

Exercises

1. For a path connected space, show that Π(X) and π1(X,x) (viewed as agroupoid) are equivalent as categories. Deduce the equivalence of (1) and(2) in theorem 13.3.1.

2. Given a locally constant sheaf F , show that F is isomorphic to the sheafassociated to a local system, and that this local system is determined upto isomorphism.

13.4. Higher direct images*

In this section, we return to general sheaf theory. Let f : X → Y be a con-tinuous map and F ∈ Sh(X) a sheaf. We can define the higher direct images byimitating the definition of Hi(X,F) in section 4.2.

R0f∗F = f∗FR1f∗F = coker[f∗G(F)→ f∗C

1(F)]Rn+1f∗F = R1f∗C

n(F)

Note that when Y is a point, Rf i∗F is just Hi(X,F) viewed as sheaf on it. Wehave an analogue of theorem 4.2.3.

Theorem 13.4.1. Given an exact sequence of sheaves

0→ A→ B → C → 0,

there is a long exact sequence

0→ R0f∗A → R0f∗B → R0f∗C → R1f∗A → . . .

There is an alternative description which is a bit more convenient.

Lemma 13.4.2. If f : X → Y is a continuous map, and F ∈ Sh(X), thenRif∗F(U) is the sheafification of the presheaf U 7→ Hi(U,F)

Proof. Let Γi denote the presheaf U 7→ Hi(U,F). For i = 0, we have Γ0 =f∗F by definition of f∗. By our original construction of H1(U,F), we have an exactsequence

f∗G(F)(U)→ f∗C1(F)(U)→ Γ1(U)→ 0

This shows that Γ1 is the cokernel f∗G(F)→ f∗C1(F) in the category of presheaves.

Hence (Γ1)+ = R1f∗F . The rest follows by induction

13.4. HIGHER DIRECT IMAGES* 171

Each element of Hi(X,F) determines a global section of the presheaf Γi andhence of the sheaf Rif∗F . This map Hi(X,F)→ H0(X,Rif∗F) is often called anedge homomorphism.

Let us now assume that f : X → Y is a fibre bundle of triangulable spaces.Then choosing a contractible neighbourhood U of y, we see that Hi(U,Z) ∼=Hi(Xy,Z). Since such neighbourhoods are cofinal, it follows that Rif∗Z is locallyconstant. This coincides with the sheaf associated to the local system Hi(Xy,Z)constructed in the previous section. Note that the sheaves Rif∗Z are definedwhen f is arbitrary, but they need not be locally constant. The global sectionsof H0(Y,Rif∗Z) can be identified with the space Hi(Xy,Z)π1(Y,y) of cohomologyclasses of the fibre invariant under monodromy. We can construct elements of thisspace using the edge homomorphism, which corresponds to restriction

Hi(X)→ Hi(Xy)π1(Y,y) ⊆ Hi(Xy).

The importance of the higher direct images is that they provide a mechanismfor computing the cohomology of X in terms of data on Y . More precisely, con-struct G•(F) as in section 11.4. Then as we saw Γ(G•(F)) is a complex whosecohomology groups are exactly H∗(X,F). We can factor this construction throughY , by considering the complex of sheaves f∗G•(F) on Y . We will denote this byRf∗F even though this isn’t technically quite correct1. The interesting features ofthis complex can be summarized by

Proposition 13.4.3.(1) The ith cohomology sheaf Hi(Rf∗F) ∼= Rif∗F .(2) Hi(Rf∗F) ∼= Hi(X,F)

To make the relationship between Rf∗F and Rif∗F clearer, let us use thetruncation operators introduced in §11.4. From lemma 11.4.2, we obtain.

Lemma 13.4.4.

0→ τ≤q−1Rf∗F → τ≤qRf∗F → Rqf∗F [−q]→ 0

for each q.

From here it is a straight forward matter to construct the Leray spectral se-quence

Epq2 = Hp(Y,Rqf∗F)⇒ Hp+q(X,F),We won’t do this, but see the exercises for some further hints. However, we willexplain one of its consequences.

Proposition 13.4.5. If F is a sheaf of vector spaces over a field, such that∑dimHp(Y,Rqf∗F) <∞, then

dimHi(X,F) ≤∑p+q=i

dimHp(Y,Rqf∗F)

and ∑i

dim(−1)iHi(X,F) =∑p,q

(−1)p+q dimHp(Y,Rqf∗F)

Proof. This follows from 13.4.4 an induction.

1A better approximation of its true meaning would be to consider Rf∗F as the quasi-isomorphism class of f∗G•(F).


Exercises

1. Prove theorem 13.4.1.2. Complete the proof of proposition 13.4.5. (See 11.4.4, for some hints.)3. Show that equality holds in proposition 13.4.5 if and only of τ≤• is strict.4. Let Rq = Rqf∗F . Define the map d2 : Hp(Y,Rq) → Hp+2(Y,Rq−1) as

the composition of the connecting map associated to 13.4.4, and the mapH∗(τ≤q−1)→ H∗(Rq−1[−q + 1]). This is the first step in the constructionof Leray spectral sequence. Show that d2 vanishes if τ≤• is strict.

13.5. First Betti number of a fibred variety*

We will return to geometry, and use the previous ideas to compute the firstBetti number of an elliptic surface. Suppose that f : X → C is a map of a smoothprojective variety onto a smooth projective curve. Assume that f has connectedfibres. Let S ⊂ C be the set of points for which the fibres are singular, and let Ube its complement. The map f−1U → U is a submersion and hence a fibre bundle.We have a monodromy representation of π1(U) on the cohomology of the fibre.

Theorem 13.5.1. b1(C) ≤ b1(X) ≤ b1(C) + dimH1(Xy,Q)π1(U)

The proof will be carried out in a series of steps. An analytic hypersurface of acomplex manifold is a subset which is locally the zero set of a holomorphic function.

Lemma 13.5.2. Let S ⊂W be an analytic hypersurface of a complex manifold,then H1(W,Q)→ H1(W − S,Q) is injective.

Proof. We prove this under the extra assumption that W has finite topolog-ical type. Then by using Mayer-Vietoris and induction, we can reduce to the casewhere W is a ball. The result is obvious in this case.

Let j : U → C denote the inclusion, and let F = R1f∗Q. There is a canonicalmorphism F → j∗j

∗F induced by the restriction maps F(V ) → F(V ∩ U), as Vranges over open subsets of C.

Corollary 13.5.3. F → j∗j∗F is a monomorphism.

Proof. This follows from the injectivity of the restriction maps H1(V,Q) →H1(V ∩ U,Q).

Proof of theorem. Proposition 13.4.5 gives

b1(X) ≤ dimH1(C, f∗Q) + dimH0(C,R1f∗Q)

Since the fibres of f are connected, f∗Q is easily seen to be Q. So the dimension ofits first cohomology is just b1(C). The previous corollary implies that

H0(C,R1f∗Q)→ H0(C, j∗j∗F) = H0(U,R1f∗Q) = H1(Xy,Q)π1(U)

is injective. This proves the upper inequality

b1(X) ≤ b1(C) + dimH1(Xy,Q)π1(U)

For the lower inequality, note that a nonzero holomorphic 1-form on C willpullback to a nonzero form on X. Thus h10(C) ≤ h10(X).

Corollary 13.5.4. If E → C is an elliptic surface such that monodromy isnontrivial, then b1(E) = b1(C).

13.5. FIRST BETTI NUMBER OF A FIBRED VARIETY* 173

Proof. By assumption, dimH1(Ey)π1(U) ≤ 1. Therefore b1(C) ≤ b1(X) ≤b1(C) + 1. Since b1(X) is even, this forces b1(E) = b1(C).

Exercises

1. Let G be a finite group of automorphisms of a curve C. Suppose thatG acts on a smooth projective variety F and therefore on C × F by thediagonal action. Let f : X = (C × F )/G → C = C/G be the projection.Check that b1(X) = b1(C) + dimH1(Xy,Q)π1(C).

CHAPTER 14

The Hard Lefschetz Theorem

In this chapter we discuss the hard Lefschetz theorem which is a structuretheorem for the cohomology of a compact Kahler manifold. Lefschetz stated aversion of this theorem for algebraic varieties in his book [L] with an incompleteproof. The first correct proof was due to Hodge using harmonic forms. A subsequentarithmetic proof for varieties was given by Deligne [D5]. At present, there does notappear to be a purely topological proof. Like the Hodge decomposition theorem, thehard Lefschetz has a number implications for the topology of projective manifoldsand more generally compact Kahler manifolds. We discuss a number of these.

14.1. Hard Lefschetz

Let X be an n dimensional compact Kahler manifold. Recall that L was definedby wedging (or more correctly cupping) with the Kahler class [ω]. The space

P i(X) = ker[Ln−i+1 : Hi(X,C)→ H2n−i+2(X,C)]

is called the primitive cohomology.

Theorem 14.1.1 (Hard Lefschetz). For every i,

Li : Hn−i(X,C)→ Hn+i(X,C)

is an isomorphism. For every i,

Hi(X,C) =[i/2]⊕j=0

LjP i−2j(X)

We indicate the proof in the next section. As a simple corollary we find thatthe Betti numbers bn−i = bn+i. Of course, this is nothing new since this also followsPoincare duality. However, it is easy to extract some less trivial “numerology”.

Corollary 14.1.2. The Betti numbers satisfy bi−2 ≤ bi for i ≤ n/2.

Proof. The theorem implies that the map L : Hi−2(X)→ Hi(X) is injective.

Suppose that X = Pn with the Fubini-Study metric 10.1.5. the Kahler classω = c1(O(1)). The class c1(O(1))i 6= 0 is the fundamental class of a codimensioni linear space (see sections 7.2 and 7.5), so it is nonzero. Since all the cohomologygroups of Pn are either 0 or 1 dimensional, this implies the Hard Lefschetz theoremfor Pn. Things get much more interesting when X ⊂ Pn is a nonsingular subvari-ety with induced metric. By Poinare duality and the previous remarks, we get astatement closer to what Lefschetz would have stated. Namely, that any element

175

176 14. THE HARD LEFSCHETZ THEOREM

of Hn−i(X,Q) is homologous to the intersection of a class in Hn+i(X,Q) with acodimension i subspace.

The Hodge index theorem for surfaces generalizes to a set of inequalities, calledthe Hodge-Riemann bilinear relations, to an dimensional compact Kahler manifoldX. Consider the pairing

Hi(X,C)×Hi(X,C)→ Cdefined by

Q(α, β) = (−1)i(i−1)/2

∫X

α ∧ β ∧ ωn−i

Theorem 14.1.3. Hi(X) = ⊕Hpq(X) is an orthogonal decomposition withrespect to Q. If α ∈ P p+q(X) ∩Hpq(X) is nonzero,

√−1

p−qQ(α, α) > 0.

Proof. See [GH, p.123].

We define the Weil operator C : Hi(X) → Hi(X) which acts on Hpq(X) bymultiplication by

√−1

p−q.

Corollary 14.1.4. The form Q(α, β) = Q(α,Cβ) on P i(X) is positive defi-nite Hermitean.

Exercises

1. When X is compact Kahler, show that Q gives a nondegenerate skew sym-metric pairing on P i(X) with i odd. Use this to another proof that bi(X)is even.

2. Determine which products of spheres Sn×Sm can admit Kahler structures.

14.2. Proof of Hard Lefschetz

Let X be as in the previous section. We defined the operators L,Λ acting formsE•(X) in section 10.1. We define a new operator H which acts by multiplicationby n− i on E i(X). Then:

Proposition 14.2.1. The following hold:(1) [Λ, L] = H(2) [H,L] = −2L(3) [H,Λ] = 2Λ

Furthermore these operators commute with ∆.

Proof. See [GH, p 115, 121].

This proposition plus the following theorem of linear algebra will prove theHard Lefschetz theorem.

Theorem 14.2.2. Let V be a vector space with endomorphisms L,Λ, H satis-fying the above identities. Then

(a) H is diagonalizable with integer eigenvalues.(b) For each a ∈ Z let Va be the space of eigenvectors of H with eigenvalue a.

Then Li induces an isomorphism between Vi and V−i.

14.2. PROOF OF HARD LEFSCHETZ 177

(c) If P = ker(Λ), then

V = P ⊕ LP ⊕ L2P ⊕ . . .(d) α ∈ P ∩ Vi then Li+1α = 0.

We give the main ideas of the proof. Consider Lie algebra sl2(C) of space oftraceless 2× 2 complex matrices. This is a Lie algebra with a basis given by

λ =(

0 10 0

)

` =(

0 01 0

)h =

(1 00 −1

)These matrices satisfy

[λ, `] = h, [h, λ] = 2λ, [h, `] = −2`

So the hypothesis of the theorem is simply that the linear map sl2(C) → End(V )determined by λ 7→ Λ, ` 7→ L, h 7→ H, preserves the bracket. Or equivalentlythat V is a representation of sl2(C). The theorem can then be deduced from thefollowing two facts from representation theory of sl2(C) (which can be found inalmost any book on Lie theory, e.g. [FH] ):

1. Every representation of sl2(C) is a direct sum of irreducible representations.Irreducibility means that the representation contains no proper nonzerosubrepresentations, i.e. subspaces stable under the action of sl2(C).

2. There is a unique irreducible representation of dimension N + 1, for eachN ≥ 0, namely SN (C2) where C2 is the standard representation. Let e1 =(1, 0)T and e2 = (0, 1)T be the standard basis of C2. Then eN1 , e

N−11 e2, . . . e

N2

gives a basis of SN (C2) and the operators act by

λ(ei1ej2) = jei+1

1 ej−12

`(ei1ej2) = iei−1

1 ej+12

h(ei1ej2) = (i− j)ei1e

j2

It suffices to check the theorem when V = SN (C2). In this case, we see thatthe span of ei1e

N−i2 is precisely the eigenspace V2i−N of h, and all eigenspaces are

of this form. The operators `, λ shift these eigenspaces as pictured

0 VN ∼= C`∼= //

λoo VN−2

∼= Cλ∼=

oo`∼= // . . .λ∼=oo

`∼= //V−N ∼= C

λ∼=oo

` // 0

This implies (b). For the remaining properties, we have that P = ker(λ) is the spanof eN1 . Thus

V = P ⊕ `P ⊕ `2P . . .as expected.

Exercises

1. Using the relations, check that λ(Va) ⊆ Va+2 and `(Va) ⊆ Va−2 for anyrepresentation V .


2. If V is a representation and v ∈ ker(λ), show that v, `v, `2, . . . spans asubrepresentation of V . In particular, this spans V if it is irreducible.

14.3. Barth’s theorem*

Barth proved a variation on the Lefschetz hyperplane theorem that nonsingularsubvarieties of Pn have the same cohomology in low degree. Hartshorne [Ha1] gavea short elegant proof which we reproduce here.

Lemma 14.3.1. Let Y ⊂ X be oriented submanifold of a compact oriented C∞

manifold. Then the following identities hold:(1) ι∗(β ∪ γ) = ι∗β ∪ ι∗γ(2) ι!ι

∗β = [Y ] ∪ β.(3) ι∗ι!α = i∗[Y ] ∪ α.

Proof. The first identity is simply a restatement of functoriality of cup prod-uct. The second is usually called the projection formula, and it can be checked bythe calculation: ∫

X

ι!ι∗β ∪ γ =

∫Y

ι∗β ∪ ι∗γ =∫X

[Y ] ∪ β ∪ γ

The same argument shows that (2) holds more generally for Y a compact sub-manifold of a noncompact manifold. In this case i! and [Y ] would take values inH∗c (X).

We can factor i as

Yj→ T

k→ X

where T is a tubular neighbourhood of Y . Let π : T → Y the projection. We have

ι∗ι!α = j∗k∗k!j!α = j∗j!α

where k! is extension by 0. By (2)

j∗j!α = j∗(j!j∗)π∗α = j∗[Y ] ∪ j∗π∗α = i∗[Y ] ∪ α

Theorem 14.3.2 (Barth). If Y ⊂ Pn is a nonsingular, complex projectivevariety, then Hi(Pn,Q)→ Hi(Y,Q) is an isomorphism for i ≤ 2 dimY − n.

Proof. Let X = Pn, m = dimY and let ι : Y → X denote the inclusion.Let L be the Lefschetz operator associated to H and H|Y (it will be clear fromcontext, which is which). Then H2(n−m)(X) is generated by Ln−m. Therefore[Y ] = d[H]n−m with d 6= 0. Consider the diagram

Hi(X) Ln−m//

ι∗

Hi+2(n−m)(X)

ι∗

Hi(Y ) Ln−m

//

(1/d)ι!88ppppppppppp

Hi+2(n−m)(Y )

The diagram commutes thanks to the previous lemma.The map Ln−m : Hi(X)→ Hi+2(n−m)(X) is an isomorphism, and so it follows

that the restriction ι∗ : Hi(X) → Hi(Y ) is injective. Therefore it is enough to

14.4. LEFSCHETZ PENCILS* 179

prove that they have the same dimension. Hard Lefschetz for Y implies that Ln−m :Hi(Y )→ Hi+2(n−m)(Y ) is injective. Therefore the same is true of ι!. Thus

bi(X) ≤ bi(Y ) ≤ bi+2(n−m)(X) = bi(X)

When Y has codimension one, we haveHi(Pn,Q) ∼= Hi(Y,Q) for i ≤ n−2. Thisis a special case of the Lefschetz hyperplane theorem, or weak Lefschetz theorem,which says that for any smooth X ⊂ Pn in general position with respect to Y ,

Hi(X,Z) ∼= Hi(X ∩ Y,Z)

for i ≤ dimX − 2.

Exercises

1. Check that the bound in Barth’s theorem is sharp.

14.4. Lefschetz pencils*

In this section, we explain Lefschetz’s original approach to the hard Lefschetztheorem. This involves the comparison of the topology variety with a family ofhyperplane sections. Modern references for this material are [La], [SGA7] and[V].

Let PN be the dual projective space whose points correspond to hyperplanesof PN . Given an n dimensional smooth projective variety X, Bertini’s theorem[Har] says that the set of hyperplanes in PN for which X ∩H is singular forms aproper Zariski closed set. Pick H “general” so that it lies in the complement of thisset, and let Y = X ∩ H. Lefschetz’s hyperplane theorem yields an isomorphismH∗(X) ∼= H∗(Y ) in low degrees. So we look at the first case where these aredifferent. We define

I = im[ι∗ : Hn−1(X,Q)→ Hn−1(Y,Q)],

andV = ker[ι! : Hn−1(Y,Q)→ Hn+1(X,Q)]

where ι : Y → X is the inclusion. We will see toward the end of this section, thatthese are spaces of vanishing cycles and invariant cycles respectively.

Proposition 14.4.1. Hn−1(Y ) = I ⊕ VProof. The composition

Hn−1(X)→ Hn−1(Y )→ Hn+1(X)

can be identified with the Lefschetz operator L which is an isomorphism. Thereforegiven ι∗β ∈ I ∩ V , we get β = L−1ι!(β) = 0. Furthermore given α ∈ Hn−1(Y ),

α = ι∗L−1ι!(α) + (α− ι∗L−1ι!(α))

decomposes it into an element of I + V .

Lefschetz’s original formulation was essentially a homological version of thisproposition. We explain the main ideas. We can and will assume that X ⊂ PN isnondegenerate which means that X does note lie on a hyperplane. The dual variety

X = H ∈ PN | H contains a tangent space of Xparameterizes hyperplanes such that H ∩X is singular.


Proposition 14.4.2. If H ∈ X is a smooth point, then H ∩X has exactly onesingular point which is a node (i.e. the completed local ring of the singularity isisomorphic to C[[x1, . . . xn]]/(x2

1 + . . . x2n)).

Proof. [SGA7, chap XVII].

A line Htt∈P1 ⊂ PN is called a pencil of hyperplanes. Any two hyperplanes ofthe pencil will intersect in a common linear subspace call the base locus. A pencilHt is called Lefschetz if Ht ∩X has at worst a single node for all t ∈ P1 and ifthe base locus H0 ∩H∞ is transverse to X.

Corollary 14.4.3. The set of Lefschetz pencils form a nonempty Zariski openset of the Grassmanian of lines in PN .

Proof. A general pencil will automatically satisfy the transversality condition.Furthermore, a general pencil will be disjoint from the singular set Xsing, since ithas codimension at least two in PN .

Given a pencil, we form an incidence variety X = (x, t) ∈ X × P1 | x ∈ HtThe second projection gives a map onto P1 whose fibres are intersections H ∩ X.There is a finite set S of t ∈ P1 with p−1t singular. Let U = P1−S. Choose t0 ∈ U ,set Xt0 = p−1(t0) and consider the diagram

X X

p

πoo

Xt0

ι

OOι

>>P1

Choose small disks ∆i around each ti ∈ S, and connect these by paths γi to thebase point t0 (figure 1).

t t0 1

γ1

Figure 1. Loops

The space p−1(γi ∪∆i) is homotopic to the singular fibre Xti = p−1(ti).

Theorem 14.4.4. Let n = dimX. Then

Hk(Xt0 ,Q)→ Hk(p−1(γi ∪∆i),Q)

is an isomorphism if k 6= n− 1, and it is surjective with a one dimensional kernelif k = n− 1.

14.4. LEFSCHETZ PENCILS* 181

Proof. Since p−1(γi ∪∆i) is homotopic to p−1(∆i), we can assume that t0 isa point on the boundary of ∆i. Let yi ∈ Xti be the singular point. We can choosecoordinates so that about yi p is given by z2

1 + z22 + . . . z2

n. Pick 0 < ρ < ε 1, andlet

B = (z1, . . . zn) | |z1|2 + |z2|2 + . . . |zn|2 ≤ ε, |z21 + z2

2 + . . . z2n| ≤ ρ

We assume, after shrinking ∆i if necessary, that ∆i is the disk of radius√ρ and

identify t0 with√ρ. If x = Re(z), y = Im(z), then we can identify

Xt0 ∩B = (x, y) ∈ Rn × Rn | ||x||2 + ||y||2 ≤ ε, ||x||2 − ||y||2 = ρ, 〈x, y〉 = 0

These inequalities imply that ||x|| 6= 0 and ||y||2 ≤ ε−ρ2 . Therefore (x, y) 7→

( x||x|| ,

2yε−ρ ) gives a homeomorphism

Xt0 ∩B ∼= (x, y) ∈ Rn × Rn | ||x||2 = 1, ||y||2 ≤ 1, 〈x, y〉 = 0

The latter space deformation retracts onto the sphere Sn−1 = (x, 0) | ||x|| = 1.It follows that Hk(Xt0 ∩ B) is generated by the fundamental class of Sn−1 whenk = n − 1 and is zero for all other k > 0. By Poincare duality it follows thatHn−1c (Xt0 ∩ B) ∼= Hn−1(Xt0 ∩ B) 6= 0. Thus we can find a form α with compact

support in Xt0 ∩B such ∫Sn−1

α 6= 0

Therefore the fundamental class δi of Sn−1 in Xt0 is nonzero.To conclude we need to appeal to a refinement of Ehresmann’s fibration theorem

[La], which will imply that X − B′ → ∆ is trivial as a bundle of manifolds withboundary, where B′ is the interior of B. Therefore the pairs (Xt0 , ∂(Xt0 −B′)) and(X − B′, ∂(X − B′)) are homotopy equivalent. Thus, by excision and the exactsequence for a pair [Hat, Sp], we have a commutative diagram

Hk+1(Xt0 , Xt0 ∩B) → Hk(Xt0 ∩B) → Hk(Xt0) → Hk(Xt0 , Xt0 ∩B)‖ ↓ ↓ ‖

Hk+1(p−1(∆i, B) → Hk(B) = 0 → Hk(X) → Hk(p−1∆i, B)

A straight forward diagram chase shows that Hk(Xt0) → Hk(X) is either an iso-morphism or injective with kernel spanned by δi when k = n− 1.

Let δi ∈ Hn−1(Xt0) be the homology class of Sn−1 constructed above. It iscalled the vanishing cycle. Let µi : Hk(Xt0) → Hk(Xt0) denote the monodromyalong the loop given by γi(∂∆i)γ−1

i . The problem of computing this can be localizedto a neighbourhood of the singularity. Let B be as in the above proof. Then excisiongives an isomorphism

Hk(Xt0 , Xt0 −B) ∼= Hk(Xt0 ∩B, ∂(Xt0 ∩B))

Note that µi acts trivially on the boundary. Thus the difference µi(α) − α is anabsolute cycle defining a class vari(α) ∈ Hk(Xt0). This defines a map

vari : Hk(Xt0 , Xt0 −B)→ Hk(Xt0)

such that the monodromy is given by

µi = I + vari can


where can : Hk(Xt0)→ Hk(Xt0 , Xt0 − B) is the natural map. It follows from thisthat

µi(α) =

α+ c(α)δi if k = n− 1α otherwise

for some constant c(α). The precise determination of the constant is given by thePicard-Lefschetz formula. We state it in cohomology which is more convenient forour purposes. We can identify the vanishing cycle with an element of cohomologyvia the Poincare duality isomorphism Hn−1(Xt0 ,Q) ∼= Hn−1(Xt0 ,Q). Then:

Theorem 14.4.5 (Picard-Lefschetz). µi(α) = α + (−1)n(n+1)/2〈α, δi〉δi where〈, 〉 denotes the cup product pairing on Hn−1(Xt0 ,Q)

Proof. See [La] or [Ar, sect 2.4].

Let V an ⊂ Hn−1(Xt0 ,Q) be the subspace spanned by all of the vanishing cyclesδi. Let Inv = Hn−1(Xt0)π1(U) be the space of of classes invariant under π1(U).

Corollary 14.4.6. The orthogonal complement V an⊥ coincides with Inv.

Proof. µi(α) = α if and only if 〈a, δi〉 = 0.

We now identify Y = Xt0 , and compare these spaces with the ones introducedat the beginning.

Proposition 14.4.7. V = V an.

Proof. The proof is adapted from [La]. Utilizing the Poincare duality iso-morphism Hn−1(Y ) ∼= Hn−1(Y ), we can work in homology. We note that underthis isomorphism, the Gysin map ι! becomes the natural (pushforward) map in ho-mology Hn−1(Y )→ Hn−1(X). Therefore its kernel can be identified with V . Thelong exact sequence for a pair shows that

V = im[Hn(X,Y )→Hn−1(Y )]

We divide P1 into upper and lower hemispheres U+ and U−. Without lossof generality, we can assume that all the singular fibres Xti of p : X → P1 arecontained in X+ = p−1(U+), and t0 lies on the boundary of U+. Let ∆i be disjointdisks around ti contained in U+. Let ri ∈ ∂∆i. Join these by paths γi. LetZ = ∪p−1γi. This retracts to Xt0 . The diagram:

Hn(X+, Xt0) → Hn−1(Xt0)‖ ‖

Hn(⋃mi=1 p

−1∆i, Z) → Hn−1(Z) ∼= Hn−1(Xt0) →⊕m

i=1 Hn−1(p−1∆i)

shows that V an is the image of Hn(X+, Xt0) in Hn−1(Xt0).Now consider the diagram

Hn(X+, Xt0) → Hn−1(Xt0)f ↓ ‖

Hn(X,Y ) → Hn−1(Y )

The map f is surjective [La, 3.6.5], and this implies that V = V an.

Proposition 14.4.8. I = Inv

14.5. DEGENERATION OF LERAY* 183

Proof. We prove this when n is odd which implies that 〈, 〉 is a nondegeneratesymmetric pairing on Hn−1(Y ). Then by corollary 14.4.6.

dim Inv = dimHn−1(Y )− dimV an

The image of Hn−1(X,Q) lies in Inv = Hn−1(Xt0)π1(U) since it factors throughHn−1(X,Q) and Hn−1(X,Q) → Hn−1(Xt0) is the edge homomorphism, see sec-tion 13.4. Thus it suffices to prove that I and inv have the dimension. But this isa consequence of propositions 14.4.1 and 14.4.7 and the previous equation:

dim I = dimHn−1(Y )− dimV = dim Inv.

We finally state the following for later use.

Theorem 14.4.9.(a) The restriction of the cup product pairing to I is nondegenerate.(b) V is an irreducible π1(U)-module, i.e. it has no π1(U)-submodules other

than 0 or V .(c) Hn−1(Y ) is a semisimple π1(U)-module (i.e. a direct sum of simple mod-

ules).

Proof. The pairing on I = imHn−1(X) given by

〈ι∗α, ι∗β〉 =∫Xt0

ι∗α ∪ ι∗β = ±Q(α, β)

is nondegenerate by corollary 14.1.4. This proves (a).We prove (b) modulo the basic fact that any two vanishing cycles are conjugate

up to sign under the action of π1(U) [La, 7.3.5]. Suppose that W ⊂ V is a π1(U)submodule, let w ∈W be a nonzero element. By (a), 〈w, δi〉 6= 0 for some i. Then

µi(w)− w = ±〈w, δi〉δi ∈W

implies that W contains δi, and thus all the vanishing cycles by the above fact.Therefore W = V .

By proposition 14.4.1, Hn−1(Y ) = I ⊕ V . The group π1(U) acts trivially on Iby proposition 14.4.8 and irreducibly on V . This implies (c).

14.5. Degeneration of Leray*

We want to mention one last consequence of the Hard Lefschetz theorem dueto Deligne [D1]. A morphism f : X → Y of complex algebraic varieties is calledprojective if it factors through a projection Y ×PN → Y . When X,Y are nonsingu-lar, the morphism f is smooth if the induced maps on algebraic tangent spaces aresurjective. (See [Har] for the definition of smoothness in general.) Ehresmann’stheorem implies that smooth projective maps are C∞ fibre bundles.

Theorem 14.5.1. Let f : X → Y be smooth projective map of smooth com-plex algebraic varieties, then the inequalities in proposition 13.4.5 for F = Q areequalities, i.e.

dimHi(X,Q) =∑p+q=i

dimHp(Y,Rqf∗Q)


Proof. We sketch the basic idea. See [GH, pp. 462-468] for further de-tails. In the exercises of section 13.4, we constructed a map d2 : Hp(Y,Rq) →Hp+2(Y,Rq−1), where Rq = Rqf∗Q, The vanishing of d2 and the higher differen-tials (which we have not constructed) is equivalent to the conclusion of the theorem.We check this for d2 only.

Let n be the dimension of the fibres. By assumption, there is an inclusionX → PN × Y which gives Lefschetz operators on the fibres. For each y, we have acorresponding Lefschetz decomposition

Hi(Xy,Q) =[i/2]⊕j=0

LjP i−2j(Xy)

In fact, we get a decomposition of sheaves

Li : Rn−i ∼= Rn+i

Ri =[i/2]⊕j=0

LjP i−2j

whereP i = ker[Ln−i+1 : Ri → R2n−i+2]

This allows us to decompose

Hp(Y,Rq) ∼=⊕

Hp(Y, P q−2j)

Thus it suffices to check vanishing of the restrictions of d2 to these factors. Considerthe diagram

Hp(Y, Pn−k)d2 //

Lk+1

Hp+2(Y,Rn−k−1)

Lk+1

Hp(Y, Pn+k+2)

d2 // Hp+2(Y,Rn+k+1)

The first vertical arrow is zero by the definition of P , and the second vertical arrowis an isomorphism by hard Lefschetz. Therefore the top d2 vanishes.

Corollary 14.5.2. If the monodromy action of π1(Y, y) on the cohomology ofthe fibre Xy is trivial (e.g. if Y is simply connected) then

dimHi(X,Q) =∑p+q=i

dimHp(Y )⊗Hq(Xy)

Exercises

1. Let π : C2 − 0 → P1C be the usual projection. This is smooth and in fact

a Zariski locally trivial C∗-bundle. The restriction S3 → P1C to the unit

sphere is called the Hopf fibration. Show that the conclusion of corollary14.5.2 fails for π and the Hopf fibration.

2. Show that b1(X) = b1(C)+dimH1(Xy,Q)π1(C) when f : X → C is smoothand projective.

14.6. POSITIVITY OF HIGHER CHERN CLASSES* 185

14.6. Positivity of higher Chern classes*

A projective morphism p : P → X of varieties is said to be a Brauer-Severimorphism if it is smooth and all the fibres are isomorphic to projective space. Thecohomology is easy to analyze using the previous results.

Lemma 14.6.1. If π : P → X is Brauer-Severi with n dimensional fibres then

Hi(P,Q) ∼=n⊕j=0

Hi−2j(X,Q)

Proof. Choose an embedding ι : P → PN × X which commutes with theprojection. Then H2(PN ,Q) = Q maps nontrivially and therefore isomorphicallyto H2(Px,Q) = Q. Since H2(Px) generates H∗(Px). The monodromy action istrivial. Therefore the lemma follow from corollary 14.5.2

Note that the proof gives a somewhat stronger result that H∗(P,Q) is generatedas an algebra by H∗(P,Q) and H2(PN ,Q). If h denotes the positive generatorc1(ι∗OPN (1)) of the latter space, then in fact

Hi(P,Q) =n⊕j=0

Hi−2j(X,Q) ∪ hj

The Brauer-Severi morphisms of interest to us arise as follows. Let E be a rankn+ 1 algebraic vector bundle on X in the original geometric sense (as opposed toa locally free sheaf). Define

P(E) = ` | ` a line in some Ex =⋃

P(Ex)

This comes with a projection π : P(E) → X, which sends ` to x. The fibre isprecisely P(Ex). The map π is projective, and there carries a class h as above.This can be constructed directly as follows. Let gij be a 1-cocycle for E withrespect to a cover Ui. Then we construct P(E) by gluing (x, [v]) ∈ Ui × Pn to(x, [gijv]) ∈ Uj × Pn. Consider the line bundle

L = (v, `) | v ∈ `, ` a line in some Exon P(E). We defineOP(E)(−1) as the sheaf of its sections, andOP(E)(1) = OP(E)(−1)∗.This need not be ι∗OPN (1) as above, however a power will. Therefore their firstChern classes will be proportional. Therefore:

Corollary 14.6.2. The above decomposition is valid for X = P(E) and h =c1(OP(E)(1)).

Remark 14.6.3. It is possible to give a much more elementary proof using theLeray-Hirsch theorem [Hat, thm 4D.1]. This has the advantage of working over theintegers and for complex C∞ vector bundles.

Although the additive structure is now determined, the multiplicative structureis more subtle. If E = On+1

Y is trivial, then P(E) = Pn ×X. Therefore hn+1 = 0by Kunneth’s formula. In general, it can be nontrivial. We can express hn+1 as alinear combination of 1, h, . . . hn with coefficients in H∗(X). These coefficients are,by definition, the Chern classes of E. More precisely, we define the ith Chern classci(E) ∈ H2i(X,Q) by

hn+1 − c1(E) ∪ hn + . . .+ (−1)n+1cn+1(E) = 0(14.6.1)


By the above remark, these classes can be defined in H2i(Y,Z). This methodof defining Chern classes, which is due to Grothendieck [GrC], is one of many.Regardless of the method, the key point is the following characterization.

Theorem 14.6.4. Chern classes satisfy the following properties:(1) For a line bundle L, c1(L) is given by the connecting map Pic(X) →

H2(X,Z) and ci(L) = 0 for i > 1.(2) The classes are functorial, i.e. ci(f∗E) = f∗ci(E) for any map f : Y → X.(3) If 0→ E1 → E → E2 → 0 is an exact sequence of vector bundles

1 + c1(E) + c2(E) + . . . = (1 + c1(E1) + . . .) ∪ (1 + c1(E2) + . . .)

Moreover, any assignment of cohomology classes to vector bundles satisfy-ing these properties coincides the theory of Chern classes.

Sketch. For the first property, note that when L is a line bundle, P(L) = Xand OP(E)(1) = L. The second is more or less immediate from the construction. Aspecial case of (3) will be given in the exercises.

The uniqueness is based on the splitting principle [F1, §3.2], which says givenX,E, there exists a map f : Y → X which induces an injection in cohomology suchthat f∗E is a successive extension of line bundles Li. Then (3) will imply that thepullback of the Chern classes are determined by c1(Li).

We want to end this chapter one last application of Hard Lefschetz. A vectorbundle E is called negative if the OP(E)(1) is ample, and E is ample if E∗ is negative.We should point that our definition of P(E) is dual the one given in [EGA, Har],so our definitions appear to be opposite of the usual ones. Also our signs for theChern classes have been adjusted accordingly.

Theorem 14.6.5 (Bloch-Gieseker). If E is a negative vector bundle of rankn+ 1 on a smooth projective variety X with d = dimX ≤ n+ 1. Then cd(E) 6= 0.

Proof. Let h = c1(OP(E)(1)), and let

η = hd−1 + c1(E)hd−1 + . . . cd−1(E) ∈ H2d−2(P(E))

Since OP(E)(1) is ample, the Hard Lefschetz theorem guarantees that

hn+2−d∪ : H2d−2(P(E))→ H2n+2(P(E))

is injective. We haveη ∪ hn+2−d = −cd(E) ∪ hn+1−d

by (14.6.1) and the fact that ci(E) = 0 for i > d. Therefore cd(E) cannot vanish.

Exercises

1. Show that there is an epimorphism π∗E → OP(E)(1).2. Let E = L1 ⊕ L2 be a sum of line bundle.

(a) Show that the divisors Zi ⊂ P(E) defined by the sections σi ∈ H0(OP(E)(1)⊗L−1i ) corresponding to the the maps π∗Li → π∗E → OP(E)(1) are dis-

joint.(b) Deduce that the product [Z1] ∪ [Z2] = 0.

14.6. POSITIVITY OF HIGHER CHERN CLASSES* 187

(c) Conclude (3) holds for E = L1⊕L2, i.e. that c1(E) = c1(L1) + c1(L2)and c2(E) = c1(L1) ∪ c1(L2).

3. Let G = G(2, n) and F = (x, `) ∈ Pn−1 × G | x ∈ `. Show that bothprojections Pn−1 ← F → G are Brauer-Severi morphisms. Use this tocalculate the Betti numbers of G.

4. F above can be identitified with P(V ), where V is the vector constructedin the exercises of §2.6. With the help of the previous exercise show thatthe cohomology of H2(G(2, 4)) is spanned by c1(V )2 and c2(V ).

5. The obstruction for a Brauer-Severi morphism to be given by P(E), forsome vector bundle E, lies in the Brauer group H2(Y,O∗Y ) (this is the ob-struction for lifting a class from H1(Y, PGLn(OY )) to H1(Y,GLn+1(OY ))).Show that the Brauer group vanishes when Y is a Riemann surface. How-ever, the obstruction can be nontrivial in general.

Part 4

Coherent Cohomology

CHAPTER 15

Coherent sheaves on Projective Space

In this chapter, we develop some algebraic tools for studying sheaves on pro-jective spaces. Most of this is done over an arbitrary field (because we can). Inthe last section, we compare the algebraic and analytic points of view over the fieldof complex numbers. The ideas in this chapter were introduced by Serre in twolandmark papers, usually abreviated as FAC [S1] and GAGA [S2].

15.1. Cohomology of line bundles on Pn

Let k be a field, and let (Pnk ,OPn) denote projective space over k viewed as analgebraic variety over k 2.3 Then Pn = Pnk has a cover by n + 1 open affine setsUi = xi 6= 0, where xi are the homogenous coordinates. Ui can be identified withaffine n-space with coordinates

x0

xi, . . .

xixi. . .

xnxi

In particular, O(Ui) is the polynomial ring in these variables. Our goal is to computecohomology of the sheavesOPn(d) using the Cech complex with respect to this cover.We first observe that this a Leray cover. Since the intersection of a finite numberof the Ui’s is affine, this will follow from:

Theorem 15.1.1 (Serre). If X is an affine variety, then

Hi(X,OX) = 0

for all i > 0.

We postpone the proof, since we will give a more general form in theorem 17.3.5.Let S = k[x0, . . . xn], and et Si ⊂ S be the space of homogeneous polynomials

of degree i. It is convenient to set Si = 0 if i < 0. Let us say that a ratio oftwo homogeneous polynomials p/q ∈ K = k(x0, . . . xn) is homogeneous of degreedeg p− deg q.

Theorem 15.1.2 (Serre). Thena) H0(Pn,OP(d)) ∼= Sd.b) Hi(Pn,OP(d)) = 0 if i 6∈ 0, n.c) Hn(Pn,OP(d)) ∼= S−d−n−1.

Proof. A complete proof can be found in [Har, III.5], from a somewhat moregeneral point of view. We use Cech cohomology with respect to the cover Ui.Our first task is to identify the Cech complex with a complex of polynomials. Thelocalization

S

[1xi

]=⊕d

S

[1xi

]d

⊂ K

191

192 15. COHERENT SHEAVES ON PROJECTIVE SPACE

has a natural Z-grading, where the dth component consists of homogeneous ele-ments of degree d. Note the that degree 0 piece of S[1/xi] is exactly k[x0/xi, . . . xn/xi] =O(Ui). Similar statements apply to localizations S[1/xi1xi2 . . .] along monomials.

We start with the description of O(d) given in section 7.3, where we identifya section of O(d)(U) with collection of rational functions fi ∈ O(U ∩ Ui) ⊂ Ksatisfying

fi =(xjxi

)dfj

or equivalently

(15.1.1) xdi fi = xdjfj

Now identify O(d)(Ui) with S[1/xi]d under the bijection which sends fi to xdi fi.Equation (15.1.1) then simply says these elements agree in K. By carrying outsimilar identifications for elements of O(d)(Ui ∩ Uj) etcetera, we can realize theCech complex for O(d) as the degree d piece of the complex

(15.1.2)⊕i

S

[1xi

]→⊕i<j

S

[1

xixj

]→ . . .

where the differentials are defined as alternating sums of the inclusions along thelines of section 7.3.

An element ofH0(Pn,O(d)) can be represented by a (n+1)-tuple (p0/xN0 , p1/x

N1 , . . .)

of homogeneous rational expressions of degree d, where the pi’s are polynomials,satisfying

p0

xN0=

p1

xN1=

p2

xN2. . .

This forces the polynomials pi to be divisible by xNi . Thus the H0(Pn,O(d)) canbe identified with Sd as claimed in (a).

From (15.1.2),⊕

dHn(Pn,O(d)) is isomorphic to S[ 1

x0...xn] modulo the the

space of coboundaries

B =∑i

S

[1

x0 . . . xi . . . xn

]S[ 1

x0...xn] is spanned by monomials xi00 . . . xinn with arbitrary integer exponents.

The image B is the space spanned by those monomials where at least one of theexponents is nonnegative. Therefore the quotient can be identified with the comple-mentary submodule spanned by monomials with negative exponents. In particular

Hn(Pn,O(d)) ∼=⊕

i0+...in=d;i1,...in<0

kxi00 . . . xinn .

This is isomorphic to S−d−n−1 via

xi00 . . . xinn 7→ x−i0−10 . . . x−in−1

n

This proves (c).It remains to prove (b). Part of this is easy. Since (15.1.2) has length n,

Hi(Pn,O(d)) is automatically zero when i > n Thus it suffices to treat 0 < i < n.We will be content to work this out for n = 2. The general case can be found

15.2. COHERENCE 193

in [Har, pp 227-228]. Our treatment will be elementary but a little ad hoc. A 1-cocycle is given by a triple pij/(xixj)e of rational expressions of degree d satisfying

p02

(x0x2)e=

p01

(x0x1)e+

p12

(x1x2)eor

(15.1.3) xe1p02 = xe2p01 + xe0p12

To show that ⊕H1(P2,O(d)) vanishes, it suffices to find polynomials qi sat-isfying

pij(xixj)e

=qixei− qjxej

or

(15.1.4) pij = xejqi − xei qjAfter expanding pij =

∑fij,k(x0, x1)xk2 and substituting into

xe2p01 = xe1p02 − xe0p12,

we can cancel xe2 to obtainp01 = xe1q0 − xe0q1

for some polynomials q0, q1. Similarly, there exists Qi such that

p12 = xe2Q1 − xe1Q2

Substituting these back into (15.1.3) shows that q1−Q1 = xe1s for some polynomials. Then q0, q1 together with q2 = Q2 + xe2s will give a solution to (15.1.4).

Exercises

1. ComputeHi(P1,OP1(d)) directly using Mayer-Vietoris with respect to U0, U1without refering to above arguments.

2. Generalize the above argument to prove that H1(Pn,OPn(d)) = 0.

15.2. Coherence

Let (X,R) be a ringed space. We need to isolate a class of R-modules withgood finiteness properties.

Definition 15.2.1. Given a ringed space (X,R), an R-module E is coherentif and only if it is locally finitely presented. This means:

(1) E is locally finitely generated, i. e., each point has a neighbourhood U suchthat R|nU surjects onto E|U for some n <∞.

(2) If R|nU → E|U is a surjection, the kernel is finitely generated.

Standard properties of coherent sheaves on ringed spaces can be found in [EGA](the definition given in [Har] is only valid for noetherian schemes).

Proposition 15.2.2. Given a short exact sequence of R-modules

0→ E1 → E2 → E3 → 0

If any two of the Ei’s are coherent, then so is the third. The tensor product, andsheaf Hom of two coherent modules is coherent.

Proof. [EGA, chap 0, 5.3].


Corollary 15.2.3. The collection of coherent R-modules and morphisms be-tween them forms an Abelian category.

Example 15.2.4. If M is a finitely generated module over a finitely generatedalgebra R, then M is coherent on MspecR.

Proposition 15.2.5. All coherent sheaves on MspecR arise as above. Moreprecisely, the functor M → M gives an equivalence between the category of finitelygenerated R-modules and the category of coherent sheaves on MspecR. The inversefunctor is the global section functor M 7→ Γ(MspecR,M).

Proof. [Har, p. 113].

The structure sheaf R of a ringed space need not be coherent in general. How-ever, by the above remarks it is for a scheme of finite type in our sense (and aNoetherian scheme in general). In the analytic case, we have:

Theorem 15.2.6 (Oka). If (X,OX) is a complex manifold, then OX is coher-ent.

Proof. [GR, p. 59]

Corollary 15.2.7. Any locally free OX-module is coherent. The ideal sheafof submanifold Y ⊂ X is coherent.

A coherent submodule I ⊂ OX is called a coherent ideal sheaf. Define

V (I) = x ∈ X | (OX/I)x 6= 0

Lemma 15.2.8. The set V (I) is closed, and (V (I), (OX/I)|V (I)) is a scheme.

We call (V (I), (OX/I)|V (I)) the closed subscheme defined by I.We state two basic results about the cohomology of coherent sheaves in the

analytic setting. A complex manifold X is called Stein if the following 3 conditionshold

(1) Global homolomorphic functions seperate points, i.e. given x1 6= x2, thereexists f ∈ O(X) such that f(x1) 6= f(x2).

(2) For an x0 ∈ X, there exists a global holomorphic functions f1, . . . fn whichgenerate the maximal ideal at x0.

(3) X is holomorphically convex, i.e. for any sequence xi ∈ X without accumi-lation points, there exist a f ∈ O(X) such that |f(xi)| → ∞.

Examples of Stein manifolds include the ball in Cn, and closed submanifolds ofCn (hence in particular, nonsingular affine varieties).

Theorem 15.2.9 (Cartan’s theorem B). Let E be a coherent sheaf on Steinmanifold then Hi(X, E) = 0 for all i > 0.

Proof. [Hrm, 7.4.3]

Theorem 15.2.10 (Cartan-Serre). If E is a coherent sheaf on a compact com-plex manifold X then the cohomology groups Hi(X, E) are finite dimensional.

Proof. When E is a vector bundle, a proof can be given using Hodge theory[GH, W]. But in general, a different type of argument is needed [CS, GR]. Wegive an indication of the proof when i = 0. The key point is to put a topology onH0(X, E) which enables us to apply results from functional analysis. Choose finite

15.3. COHERENT SHEAVES ON Pn 195

covers Uj and Vj by complex balls with Uj ⊂ Vj . We can identify H0(X, E)with the spaces of zero Cech cocycles Z0(Uj, E)) or Z0(Vj, E)). The proof canbe broken into series of observations.

(1) Any locally compact Frechet space (a locally convex complete metrizablevector space) is finite dimensional.

(2) The space of holomorphic functions O(B) on a ball carries a natural topol-ogy which makes this a Frechet space: convergence is uniform convergenceon compact sets. A similar statement holds for sections of a coherent overB.

(3) If B1 is a ball whose closure lies in the another ball B2, restriction O(B2)→O(B1) is a compact operator, i.e. the image in O(B1) of a ball in O(B2) isprecompact. This follows from a theorem of Montel about normal familiesof holomorphic functions. This holds more generally, when O is replacedby a coherent sheaf.

(4) H0(X, E) carries a natural Frechet space structure inherited from the prod-uct topology of

∏E(Uj) or

∏E(Vj).

From all of these facts, it follows that the identity on H0(X, E), which can berealized as the restriction

Z0(Uj, E)→ Z0(Vj, E),

is a compact operator. Therefore this space is locally compact, and consequentlyfinite dimensional.

The analogous statements in the algebraic category will be proved later.

Exercises

1. Let j : U → X be proper open subset of an affine variety. Show that thesheaf

j!OU =

O(V ) if V ⊆ U0 otherwise

is an OX -module which is not isomorphic to M for any M . In particular,it is not coherent.

15.3. Coherent Sheaves on Pn

Our goal, in this section, is to describe all coherent sheaves on Pn = Pnk . LetS = k[x0, . . . xn] = ⊕Si be as in section 15.1. Let π : An+1 − 0 → Pnk be thecanonical the projection.

A Z-graded S-module is an S-module M with a decomposition M = ⊕i∈ZMi

into subspaces such that SiMj ⊂ Mi+j . If M is also finitely generated as an S-module, then each of the components Mi is finite dimensional, and Mi = 0 fori 0. Starting with a finitely generated graded module M , we can produce acoherent sheaf M on An+1, restrict it to An+1 − 0, and push this down to Pn toget an OPn -module M = π∗(M |An+1−0). This will not be coherent, but it will asum of coherent sheaves M = ⊕dM(d) that will be defined shortly.

The basic open sets for the Zariski topology are of the form π(D(f)) with fa homogenous polynomial. If f ∈ Sa, then M(D(f)) = M [1/f ] = ⊕dM [1/f ]d.has a Z-grading. Elements of M [1/f ]d, are generated by the fractions m/f i with


m ∈ Md+ai. This can be applied to S to yield a grading on the ring S[1/f ], andthe degree 0 component is a subring. For example,

S

[1xi

]0

= k

[x0

xi, . . .

xnxi

]is the coordinate ring of Ui More generaly:

Lemma 15.3.1. OPn(π(D(f))) ∼= S[

1f

]0.

Proof. We prove one direction. An element of S[1/f ]0 is a linear combinationof fractions g/f i with g homogenous of the same degree as the denominator. Inparticular, the function is

[a0, . . . an] 7→ g(a0, . . . an)f(a0, . . . an)i

is well defined and regular on π(D(f)).

For M as above, M [1/f ] is a graded S[1/f ]-module. In particular, it followsthat each M [1/f ]d is an S[1/f ]0-module.

Lemma 15.3.2. There is an OPn-module M(d) such that

M(d)(π(D(f))) = M

[1f

]d

and

M =⊕d∈Z

M(d)

Proof. We define M(d)(U) to be the submodule of M(U) consisting of sec-tions m such that its restriction lies in M [1/f ]d for every π(D(f)) ⊂ U .

A homomorphism of graded S-modules f : M → N is a module homomorphismwhich preserves the grading i.e. f(Mi) ⊆ Ni. The construction M(d) is clearlyfunctorial for such maps. We define M = M(0). Given a graded module M , wedefine M(d) to be M with the grading shifted by d (M(d))i = Mi+d Given twograded modules M,N , let us say that they are stably isomorphic or M ∼ N if andonly if ⊕

i≥i0

Mi∼=⊕i≥i0

Ni

as graded S-modules for some i0.Now suppose that M is a coherent OP-module. Define

Γ∗(M) = Γ(An+1 − 0, π∗M).

Lemma 15.3.3. Γ∗(OPn) = S

Proof. Γ∗(OPn) is the ring of regular functions on An+1−0. Such a functioncan be represented by a ratio of polynomials f/g such that g(a) 6= 0 for all a ∈An+1 − 0. This forces g to be a nonzero constant.

15.3. COHERENT SHEAVES ON Pn 197

GivenM as above, we get a natural S = Γ∗(OPn) module structure on Γ∗(M).We define a grading on it in terms of the action of the group k∗. Given t ∈ k∗, letµt denote the automorphism of An+1 − 0 induced by scalar multiplication by t.Since π µt = π, given σ ∈ Γ∗(M), we can identify

µ∗tσ ∈ Γ(An+1 − 0, µ∗tπ∗M) = Γ∗(M)

WhenM = OPn , the action is given explicitly by µ∗t (f(x0, . . . xn)) = f(tx0, . . . txn).The grading on S can be recovered from this, since Si is precisely the space ofpolynomials on which each t ∈ k∗ acts by ti. In general, the ith component Γi(M)can be defined similarly as the eigenspace of the µ∗t with eigenvalue ti. In particular,Γ0(M) consists of the space of k∗-invariant sections of π∗M, and this can beidentified with Γ(M). An similar interpretation for Γi(M) is given in the exericses.

We summarize the main properties.

Theorem 15.3.4.1. Given a finitely generated graded S-module M , the OPn-modules M(d) are

coherent.2. S(d) ∼= OPn(d).

3. M(d) ∼= M(d) ∼= M ⊗OPn(d).4. The operations M 7→ M(d) are exact functors.5. Γ∗(M) ∼M6. Γ∗(M) ∼=M.

Proof. A proof can be found in [Har, II,5] using the description of Γ∗ givenin the exercises.

Corollary 15.3.5. Every coherent OPn-module is isomorphic to M for somefinitely generated graded S-module M .

It is not always true that Γ∗(M) ∼= M as the following example shows.

Example 15.3.6. Let M be a graded module which is finite dimensional overk, so Mi = 0 for i >> 0. Then M = 0.

Example 15.3.7. Let I ⊂ S be a homogenous ideal. Then I is coincides withthe sheaf defined at the end of section 3.4. This defines a closed subscheme of Pnwhich is a subvariety if I is prime.

Theorem 15.3.8. Any coherent sheaf E on Pn fits into an exact sequence

0→ Er → Er−1 → . . . E0 → E → 0

where r ≤ n and each Ei is a sum of a finite number of line bundles OPn(j).

Proof. This is a consequence of the Hilbert syzygy theorem [E, 1.13] whichsays that any finitely generated graded S-module has a finite free graded resolutionof length at most n.

For many standard examples, it is possible to construct explicit resolutions ofthis type.


Example 15.3.9. The tangent sheaf fits into an exact sequence

0→ OPn

0BB@x0

x1

. . .

1CCA−→ OPn(1)n+1 → TPn → 0

This will be justified later.

Exercises

1. Let M(i) = M [1/xi]0 viewed as an S[1/xi]0-module. Check that M(i) isisomorphic to the restriction of M to Ui. Prove coherence of M .

2. Check that example 15.3.6 works as stated.3. Given a pair of graded S-modules M and N , define a grading on their

tensor product M ⊗N = M ⊗S N by

(M ⊗N)i =∑a+b=i

Ma ⊗Nb

Prove that(a) M ⊗ S(d) = M(d).(b) (M ⊗N)(i)

∼= M(i) ⊗S(i) N(i)

(c) M ⊗N ∼= M ⊗OPn N4. Given a coherent OPn -module M, define M(d) = M⊗O(d). Check that

Γd(M) = Γ(M(d)) (hint: µ∗t (f ⊗ g) = µ∗t (f) ⊗ µ∗t (g)). Thus Γ∗(M) =⊕dΓ(M(d)).

15.4. Cohomology of coherent sheaves

Given a coherent sheaf, E on Pn, let E(d) = E ⊗ O(d). This is again coherent,and consistent with the notation M(i) by the previous exercises.

Theorem 15.4.1. If E is a coherent sheaf on Pn, then Hi(Pn, E) is finite di-mensional for each i, and is zero for i > n. Furthermore Hi(Pn, E(d)) = 0 ford 0 and i > 0.

Proof. We prove this by induction on r, where r is the length of the shortestsyzygy i.e. resolution of the type given by theorem 15.3.8. If r = 0, we are done bytheorem 15.1.2. Suppose that the theorem holds for all r′ < r. Choose a resolution

0→ Er → Er−1 → . . . E0 → E → 0

and let E ′ = ker[E0 → E ]. We have exact sequences

0→ Er → Er−1 → . . . E1 → E ′ → 0

and0→ E ′ → E0 → E → 0

Since E ′ has syzygy of length r−1, H∗(Pn, E ′) is finite dimensional by the inductionhypothesis. Therefore the long exact sequence

. . . Hi(Pn, E0)→ Hi(Pn, E)→ Hi+1(Pn, E ′) . . .implies the finite dimensionality of Hi(Pn, E). The vanishing of Hi(Pn, E(d)) fori > 0 and d 0 can be proved by induction in similar manner. Details are left asan exercise.

15.4. COHOMOLOGY OF COHERENT SHEAVES 199

The following gives an analogue to theorem 15.2.10.

Corollary 15.4.2. If E is a coherent sheaf on a projective variety X, then thecohomology groups Hi(X, E) are finite dimensional.

Proof. Embed ι : X → Pn. Then ι∗E is a coherent OP-module, so it hasfinite dimensional cohomology. Since ι∗ is exact,

Hi(X, E) ∼= Hi(Pn, ι∗E)

The Euler characteristic of a sheaf F of k-modules on a space X is

χ(F) =∑

(−1)i dimkHi(X,F)

provided that the sum is finite. The advantage of working with the Euler charac-teristic is the following:

From lemma 12.3.2, we easily obtain:

Lemma 15.4.3. If 0 → Fr → . . .F0 → F → 0 is an exact sequence of sheaveswith finite dimensional cohomology, then

χ(F) =∑j

(−1)jχ(Fj)

Proof. Break this up into short exact sequences and use induction.

Theorem 15.4.4. If E is a coherent sheaf on Pnk , then i → χ(E(i)) is a poly-nomial in i of degree at most n. If X is a subvariety of Pnk and E = OX , then thispolynomial has degree dimX.

Proof. From theorem 15.1.2,

χ(OPn(i)) =

dimSi if i ≥ 0(−1)n dimS−d−n−1 otherwise

=(n+ i

n

),

which is a polynomial of degree n. This implies that χ(E(i)) is a polynomial ofdegree ≤ n when E is sum of line bundles OPn(j). In general, theorem 15.3.8 andthe above corollary implies that for any coherent E

χ(E(i)) =r∑j=0

(−1)jχ(Ej(i))

where Ej are sums of line bundles. This shows that χ(E(i)) is polynomial withdegree at most n.

To get the sharper bound on degree when E = OX , we use induction on dimXand the relation

χ(OH∩X(i)) = χ(OX(i))− χ(OX(i− 1))which follows from the sequence

0→ OX(−1)→ OX → OH∩X → 0,

where H is a general hyperplane.


χ(E(i)) is called the Hilbert polynomial of E . It has a elementary algebraicinterpretation

Corollary 15.4.5 (Hilbert). Let M be a finitely generated graded S-module,then i 7→ dimMi is a polynomial for i 0, and this coincides with the Hilbertpolynomial χ(M(i)).

Exercises

1. Let f be a homogeneous polynomial of degree d in S = k[x0, x1, x2]. Thencorresponding to the exact sequence of graded modules

0→ S(−d) ∼= Sf → S → S/(f)→ 0

there is an exact sequence of sheaves

0→ OP2(−d)→ OP2 → OX → 0.

Prove that

dimH1(X,OX) =(d− 1)(d− 2)

22. Let f be a homogeneous polynomial of degree d in 4 variables. Repeat the

above exercise for H1(X,OX) and H2(X,OX).3. Prove the remaining parts of theorem 15.4.1.4. if E is a nonzero coherent sheaf on Pn, prove that the degree of χ(E(i))

coincdies with the dimension of its support supp(E) = x ∈ Pn | Ex 6= 0.

15.5. GAGA

Let k = C throughout this section. Let (Pn,OPn) denote complex projectivespace as an algebraic variety, and (Pnan,OPn

an) projective space as a complex mani-

fold. In other words, Pn (resp. Pnan) is endowed with the Zariski (strong) topologyand OPn (resp. OPn

an) is the sheaf of algebraic (resp. holomorphic) functions. We

have a morphism of ringed spaces

ι : (Pnan,OPan)→ (Pn,OP)

Given an OP-module E , define the OPan-module Ean = ι∗E . Ean is coherent if E

is. This operation gives a functor between the categories of coherent OP and OPan

modules. We have the following fact, which we will need below:

Lemma 15.5.1. This functor is exact. If E 6= 0, then Ean 6= 0.

Proof. It suffices to check these statements at the stalk level. We have Ean,x =Ex⊗OPn

anxand this functor is exact, because OPn

an,xis a faithfully flat extension of

OPn,x. See [E] for a general discussion of faithful flatness, and [S2] for this specificcase.

As a warm up to the “GAGA” theorems, we work out a special case for P1.Since Hi(P1

an,OP1an

) = 0 for i > 0 by the Hodge theorem for example, the expo-nential sequence yields

c1 : H1(P1,O∗P1an

) ∼= H2(P1an,Z) = Z

Sincec1(OP1(d)an)) = c1(OP1(1)⊗dan ) = dc1(OP1(1)an) = d

15.5. GAGA 201

The set of isomorphism classes of analytic line bundles coincides with the setOP1(d)an of isomorphism classes of algebraic line bundles. Let us compare theircohomology. We can see that U0, U1 is a Leray cover for OP1(d)an by apply-ing Cartan’s theorem B (theorem 15.2.9) – of course, this is a bit of overkill. Setz = x1/x0 and identify this with a trivializing section OP1(d)an(U0) (i.e. the imageof 1 under an isomorphism OP1

an|U0∼= OP1(d)an|U0 .) Then zd extends to a trivi-

alizing section of OP1(d)an(U1). Thus the Cech complex can be identified with acomplex ∞∑

n=0

anzn

⊕

∞∑n=0

bnz−n

0@ 1−zd

1A−→

∞∑n=−∞

cnzn

where the above series converge in U0, U1 and U0 ∩ U1 respectively. From this itfollows easily that H0(OP1(d)an) can be identified with the space of polynomials ofdegree d in z, and H1(OP1(d)an) with the space spanned by z−d+1, . . . z−1. Thiscoincides with the cohomology of O(d) by theorem 15.1.2. Therefore

Hi(OP1(d)an) ∼= Hi(OP1(d))

More generally, we have the first GAGA theorem.

Theorem 15.5.2 (Serre). Let E be a coherent OPn-module. Then there is anisomorphism

Hi(Pn, E) ∼= Hi(Pnan, Ean)

Proof. We outline the proof, which involves a series of reductions to elemen-tary cases or devissage. A complete argument can be found in [S2]. The key stepsare as follows:Step 1. The theorem holds for OPn :

From theorem 15.1.2, we see that Hi(Pn,OPn) = 0 if i > 0 and C ifi = 0. We check the same result holds for Hi(Pnan,OPn,an) = H(0,i)(X).From 7.2, it follows that Hi(Pn,C) is zero if i is odd and generated by theclass of a linear subspace otherwise. It follows that the Hodge structure onHi(Pnan,C) is of type (i/2, i/2) when i is even. Therefore H(0,i)(X) is C ifi = 0 and zero otherwise.

Step 2. The theorem holds for OPn(d):Let H ⊂ Pn be a hyperplane. By induction on n, we can assume that

the result holds for H ∼= Pn−1. We have an exact sequence

0→ OPn(d− 1)→ OPn(d)→ OH(d)→ 0

which yields a diagram

. . . Hi(OP(d− 1)) → Hi(OP(d)) → Hi(OH(d)) . . .↓ ↓ ↓

Hi(OPan(d− 1)) → Hi(OPan

(d)) → Hi(OHan(d))

By the induction assumption and the five lemma, we see that the verticalmap is an isomorphism for OPn(d) if and only if it is for OPn(d− 1). Sincethe isomorphism holds for d = 0 by the previous step, it holds for all d.

Step 3. The theorem holds for any coherent E:When i > n, Hi(Pn, E) = 0 by theorem 15.4.1. For the analogous result

on the analytic side, we need to appeal to theorem 15.2.9, which implies


that the standard cover Uj of Pn by affine spaces is a Leray cover withrespect to E . This implies the vanishing of Hi(Pnan, Ean) for i > n, sincethe Cech complex for Uj is trivial in these degrees.

We can prove the remaining case by descending induction on i startingfrom i = n+ 1. We have an exact sequence

0→ R→ F → E → 0

whereR is coherent and F is a direct sum of line bundles by theorem 15.3.8.The previous step, induction and the five lemma will finish the argument.

The GAGA theorem can fail for nonprojective varieties. For example, H0(OCnan

)is the space of holomorphic functions on Cn which is much bigger than the spaceH0(OCn) of polynomials.

Let X ⊆ Pn be a nonsingular subvariety. The sheaf of algebraic p-forms ΩpX canbe viewed as a coherent OP-module. Then (ΩpX)an is the sheaf of holomorphic p-forms. In particular, (OX)an is the sheaf of holomorphic functions of the associatedcomplex manifold Xan. Thus:

Corollary 15.5.3. dimHq(X,ΩpX) coincides with the Hodge number hpq ofthe manifold Xan.

There is another part to the GAGA theorems which is even more surprising.

Theorem 15.5.4 (Serre). The functor E 7→ Ean induces an equivalence betweenthe categories of coherent OPn and OPn

anmodules. In particular, any coherent OPn

an-

module arises from an OPn-module, which is unique up to isomorphism.

Proof of theorem 15.5.4. We outline the steps. We will refer to a coher-ent OPn-module (respectively OPn

an-module) as an algebraic (respectively analytic)

sheaf.Step 1. Given a coherent analytic sheaf E on Pn, there exists a constant d0 such

that E(d) is generated by global sections for d ≥ d0:We assume for simplicity that E is torsion free (see the exercises for the

general case). The proof goes by induction on n. We note that step 2 willfollow from step 1. Thus we can assume that the statements of both stepsare valid for a hyperplane. Given x ∈ Pn, choose a hyperplane H passingthrough x. Consider the sequence

0→ OPn(d− 1)→ OPn(d)→ OH → 0

Tensoring this with E yields

0→ E(d− 1)→ E(d)→ E|H(d)→ 0

(the injectivity of the first map follows from torsion freeness of E). By step2 for H, we have that H1(E|H(d)) = 0 for d ≥ d1. Therefore, there is a d1

such that for d ≥ d1 we get a sequence of surjections

H1(E(d− 1))→ H1(E(d))→ H1(E(d+ 1)) . . .

Since these spaces are finite dimensional (theorem 15.2.10), these mapsmust stabilize to isomorphisms for all d ≥ d2 for some d2 ≥ d1. Thus

H0(E(d))→ H0(E|H(d))

15.5. GAGA 203

is surjective for d ≥ d2. By induction, we can assume that E|H(d) isgenerated by global sections H0(E(d)|H) for d ≥ d0 ≥ d2. These can belifted to sections H0(E(d)) which will span the stalk E(d)x.

Step 2. Given a coherent analytic sheaf E on Pn, there exists a constant d0 suchthat Hi(Pn, E(d)) = 0 for d ≥ d0:

Arguing as in the proof of theorem 15.5.2, we see that Hi(Pn, E(d)) = 0for i > n and any d. From step 1, we have that E(a) is generated by globalsections s1, . . . sN for some a 0. These sections can be viewed as mapsO(−a)→ E . Thus we have an exact sequence

0→ R→ FPsj−→ E → 0

where F = O(−a)N , and R is the kernel. Since, F is is a direct sum of linebundles, by theorems 15.5.2 and 15.1.2, we have Hi(F(d)) = 0 for i > 0and d 0. Therefore

Hi(E(d)) ∼= Hi+1(R(d))

Thus this step follows by descending induction on i.Step 3. If E ,F are algebraic, then Hom(E ,F) ∼= Hom(Ean,Fan):

The left and right hand sides are the spaces of global sections ofHom(F , E) and

Hom(F , E)an = Hom(Fan, Ean)

respectively. These spaces are isomorphic by theorem 15.5.2.Step 4. Any analytic sheaf E arises from an algebraic sheaf:

Arguing as in step 2, we see that E fits into an exact sequence

0→ R→ F2 → E → 0

where F2 is a direct sum of line bundles. Repeating this yields an epimor-phism

F1 → R→ 0where F1 is a direct sum of line bundles. Thus we obtain a presentation

F1f→ F2 → E → 0

In particular, note that the Fi are algebraic. It follows from the previousstep that f is a morphism of the underlying algebraic sheaves. Conse-quently E is the cokernel of an algebraic map, and hence also algebraic.

Corollary 15.5.5 (Chow’s theorem). Every complex submanifold of Pn is anonsingular projective algebraic subvariety.

Proof. The analytic ideal sheaf J of a submanifold X of Pn is a coherentOPn

an-module. Thus J = Ian, for some coherent submodule I ⊂ OPn . The subva-

riety defined by I is easily seen to coincide with X as a set.

Note that this implies the assertion we made some time ago that all compactRiemman surfaces are algebraic curves.

Corollary 15.5.6. A holomorphic map between nonsingular projective alge-braic is a morphism of varieties.

Proof. Apply Chow’s theorem to the graph of the map.


Exercises

1. IfX is a smooth projective variety, show thatH1(X,O∗X) ∼= H1(Xan,O∗Xan).

2. Show that GAGA fails for constant sheaves, by showing for any irreduciblevariety (with Zariski topology) Hi(X,ZX) = 0 for i > 0.

3. Show that a meromorphic function on smooth projective curve is the samething as a rational function. This is also true in higher dimensions, but theproof would be harder.

4. Consider the set up in as step 1 of the proof of theorem 15.5.4, but withoutthe torsion freeness assumption for E . Let K = ker[E(−1)→ E ], where themap is given by tensoring E with the inclusion of the ideal sheaf of H.(a) Show that K is supported on H. Thus we can apply the induction

assumption to conclude that Hi(K(d)) = 0 for i > 0 and d 0.(b) By breaking up

0→ K(d)→ E(d− 1)→ E(d)→ E(d)|H → 0

into short exact sequences, show that H0(E(d))→ H0(E(d)|H) is sur-jective for d 0.

CHAPTER 16

Computation of some Hodge numbers

Hodge theory produces a number of useful invariants in algebraic geometry. Butthis begs the question of how would one actually compute these in practice. In thecase of Hodge numbers, the GAGA theorem 15.5.2 allows us to do this by workingin the algebraic setting. We work this out for projective spaces, hypersurfaces andrelated examples.

16.1. Hodge numbers of Pn

Let S = k[x0, . . . xn] and P = Pnk for some field k. We first need to determinethe cotangent sheaf.

Proposition 16.1.1. There is an exact sequence

0→ Ω1P → OP(−1)n+1 → OP → 0

Proof. Let ΩS = ⊕Sdxi ∼= Sn+1 be the cotangent module of S. Constructthe graded S-module

M = Γ∗(Ω1P) = Γ(An+1 − 0, π∗Ω1

P),

where π : An+1−0 → P is the projection. This can be realized as the submoduleΩS consisting of those forms which annhilate the tangent spaces of the fibres of π.These tangent space of the fibre over [x0, . . . xn] is generated by the Euler vectorfield

∑xi

∂∂xi

. Thus a 1-form∑fidxi lies in M if and only if

∑fixi = 0.

Next, we have to check the gradings. ΩS has a grading, such that dxi lie indegree 0. Under the natural grading of M = Γ∗(Ω1

P), sections of Ω1P(Ui), which are

generated by

d

(xjxi

)=xidxj − xjdxi

x2i

should have degree 0. Thus the gradings on ΩS and M are off by a shift of one.To conclude, we have an exact sequence of graded modules

(16.1.1) 0→M → ΩS(−1)→ m→ 0

where m = (x0, . . . xn) and the first map sends dxi to xi. Since m ∼ S, (16.1.1)implies the result.

Dualizing yields 15.3.9.

Proposition 16.1.2. Suppose we are given an exact sequence of locally freesheaves

0→ A→ B → C → 0If A has rank one, then

0→ A⊗∧p−1C → ∧pB → ∧pC → 0

205

206 16. COMPUTATION OF SOME HODGE NUMBERS

is exact for any p ≥ 1. If C has rank one, then

0→ ∧pA → ∧pB → ∧p−1A⊗ C → 0

is exact for any p ≥ 1.

Proof. We prove the first statement where rk(A) = 1 by induction, leavingthe second as an exercise. When p = 1 we have the original sequence. In general,the maps in the putative exact sequence need to be explained. The last map∧pB → ∧pC is the natural one. The multiplication B ⊗ ∧p−1B → ∧pB restricts togive µ : A⊗ ∧p−1B → ∧pB. We claim that µ factors through A⊗ ∧p−1C. For thiswe can, by induction, appeal to the exactness of

0→ A⊗∧p−2C → ∧p−1B → ∧p−1C → 0

Since A has rank one,µ|A⊗(A⊗∧p−2C) = 0

so µ factors as claimed. Therefore all the maps in the sequence are defined.Exactness can be checked on stalks. Thus the sheaves can be replaced by free

modules. Let b0, b1, . . . be a basis for B with b0 spanning A. Then the imagesb1, b2, . . . give a basis for C. Then above maps are given by

b0 ⊗ bi1 ∧ . . . ∧ bip−1 7→ b0 ⊗ bi1 ∧ . . . ∧ bip−1

bi1 ∧ . . . ∧ bip 7→ bi1 ∧ . . . ∧ bipwhere . . . i2 > i1 > 0. The exactness is immediate.

Corollary 16.1.3. There is an exact sequence

0→ ΩpP → OP(−p)(n+1

p ) → Ωp−1P → 0

and in particularΩnP ∼= OP(−n− 1)

Proof. This follows from the above proposition and proposition 16.1.1, to-gether with the isomorphism

∧p[OP(−1)n+1] ∼= OP(−p)(n+1

p )

This corollary can be understood from another point of view. Using thenotation introduced in the proof of proposition 16.1.1, we can extend the mapΩS(−1)→ m to an exact sequence

(16.1.2) 0→ [∧n+1ΩS ](−n− 1) δ→ . . . [∧2ΩS ](−2) δ→ ΩS(−1) δ→ m→ 0

whereδ(dxi1 ∧ . . . dxip) =

∑(−1)pxijdxi1 ∧ . . . dxij ∧ . . . dxip

is contraction with the Euler vector field. The sequence is called the Koszul complexand it is one of the basic workhorses of homological algebra [E, chap 17]. Theassociated sequence of sheaves is

0→ [∧n+1On+1P ](−n− 1)→ . . . [∧2On+1

P ](−2)→ [On+1P ](−1)→ OP → 0

If we break this up into short exact sequences, then we obtain exactly the sequencesin corollary 16.1.3.

16.1. HODGE NUMBERS OF Pn 207

Proposition 16.1.4.

Hq(P,ΩpP) =k if p = q ≤ n0 otherwise

Proof. When p = 0 this follows from theorem 15.1.2. In general, this theoremtogether with corollary 16.1.3 implies

Hq(ΩpP) ∼= Hq−1(Ωp−1P )

so we get the result by induction.

When k = C, this gives a new proof of the formula for Betti numbers of Pngiven in section 7.2. By a somewhat more involved induction we can obtain:

Theorem 16.1.5 (Bott). Hq(Pn,ΩpPn(r)) = 0 unless

(a) p = q, r = 0(b) q = 0, r > p(c) or q = n, r < −n+ p

Proof. A complete proof will be left for the exercises. We give the proof forp ≤ 1. For p = 0, this is a consequence of theorem 15.1.2. We now turn to p = 1.Corollary 16.1.3 implies that

Hq−1(O(r))→ Hq(Ω1(r))→ Hq(O(r − 1))b

is exact. Here we simply denote the binomial coefficients by b, b′ . . ., since the exactexpressions will be immaterial. This proves that Hq(Ω1(r)) = 0 for q = 0 & r < 1,q = 1 & r > 0, 1 < q < n, or q = n & r ≥ −n + 1. The remaining cases areq = 0 & r = 1 and q = 1 & r > 0. The trick is to apply corollary 16.1.3 with othervalues of p. This yields exact sequences

Hq(O(r − 2))b → Hq(Ω1(r))→ Hq+1(Ω2(r))→ Hq+1(O(r − 2))b

Hq+1(O(r − 3))b′→ Hq+1(Ω2(r))→ Hq+2(Ω3(r))→ Hq+2(O(r − 3))b

′

. . .

leading to isomorphisms

H0(Ω1(1)) ∼= H1(Ω2(1)) ∼= . . . Hn−1(Ωn(1)) = Hn−1(O(−n)) = 0

Likewise for r > 0,

H1(Ω1(r)) ∼= H2(Ω2(r)) ∼= . . . Hn(Ωn(r)) = Hn(O(−1)) = 0

Exercises

1. Finish the proof of proposition 16.1.2.2. Given an exact sequence 0 → A → B → C → 0 of locally free sheaves,

prove that the top exterior power detB ∼= (detA) ⊗ (det C). Use this torederive the formula for ΩnPn .

3. Finish the proof of theorem 16.1.5.


16.2. Hodge numbers of a hypersurface

We now let X ⊂ Pn = P be a nonsingular hypersurface defined by a degree dpolynomial.

Proposition 16.2.1. The restriction map

Hq(Pn,ΩpP)→ Hq(X,ΩpX)

is an isomorphism when p+ q < n− 1.

As corollary, we can calculate many of the Hodge numbers of X.

Corollary 16.2.2. The Hodge numbers hpq(X) = δpq, where δpq is the Kro-necker symbol, when n− 1 6= p+ q < 2n− 2.

Proof. We give a proof when k = C. For p + q < n − 1, this follows fromthe above proposition and proposition 16.1.4. For p+ q > n− 1, this follows fromGAGA and 10.2.3.

When k = C this proposition can be deduced from theorem 14.3.2, the canonicalHodge decomposition (11.2.4) and GAGA. We give a different self contained proofwhich works for any field. We start with few key lemmas.

Lemma 16.2.3.0→ ΩpP(−d)→ ΩpP → ΩpP|X → 0

is exact. (Recall that ΩpP|X is shorthand for i∗i∗ΩpP, where i : X → P is the inclu-

sion.)

Proof. Tensor0→ OP(−d)→ OP → OX → 0

with ΩpP to get

0 // ΩpP ⊗OP(−d) //

∼=

ΩpP ⊗OP //

∼=

ΩpP ⊗OX //

∼=

0

0 // ΩpP(−d) // ΩpP // ΩpP|X // 0

The first isomorphism follows from theorem 15.3.4, the second is tautological, andthe third uses the projection formula [Har, p. 124].

Lemma 16.2.4. 0→ OX(−d)→ Ω1P|X → Ω1

X → 0 is exact.

Proof. We have a natural epimorphism Ω1P|X → Ω1

X corresponding to re-striction of 1-forms. We just have to determine the kernel. Let f be a definingpolynomial of X, and let M = Γ∗(Ω1

P) and M = Γ∗(Ω1X). We embed M as a sub-

module of ΩS(−1) as in the proof of proposition 16.1.1. In particular, the symbolsdxi have degree 1. Then ker[M/fM →M ] is a free S/(f) module generated by

df =∑i

∂f

∂xidxi

Thus it is isomorphic to S/(f)(−d).

Corollary 16.2.5. 0→ Ωp−1X (−d)→ ΩpP|X → ΩpX → 0

Proof. Apply proposition 16.1.2 to the lemma.

16.3. HODGE NUMBERS OF A HYPERSURFACE II 209

It is more convenient to prove a slightly stronger form of proposition 16.2.1.

Proposition 16.2.6. If p+ q < n− 1 then

Hq(X,ΩpX(−r)) =

Hq(Pn,ΩpP) if r = 00 if r > 0

Proof. We prove this by induction on p. For p = 0, this follows from the longexact sequences associated to

0→ OP(−d− r)→ OP(−r)→ OX(−r)→ 0

and theorem 15.1.2.In general, by induction and corollary 16.2.5 we deduce

Hq(X,ΩpX(−r)) = Hq(ΩpP(−r)|X)

for r ≥ 0 and p+ q < n− 1. Lemma 16.2.3 and theorem 16.1.5 give

Hq(ΩpP(−r)|X) =

Hq(Pn,ΩpP) if r = 00 if r > 0

Exercises

1. Using an exercise from the previous section, deduce a version of the ad-junction formula Ωn−1

X∼= OX(d− n− 1). X is as above.

2. Compute Hq(X,Ωn−1X ) for all q.

16.3. Hodge numbers of a hypersurface II

We continue the notation from the previous section. By corollary 16.2.2 themiddle Hodge numbers of X are the only things left to compute. These can beexpressed by the Euler characteristics:

Lemma 16.3.1. hp,n−1−p(X) = (−1)n−1−pχ(ΩpX) + (−1)n

We can calculate these Hodge numbers by hand using the following the recur-rence formulas

Proposition 16.3.2.(a)

χ(ΩpP(i)) =p∑j=0

(−1)j(n+ 1p− j

)(i− p+ j + n

n

)(b)

χ(OX(i)) =(i+ n

n

)−(i+ n− d

n

)(c)

χ(ΩpX(i)) = χ(ΩpP(i))− χ(ΩpP(i− d))− χ(Ωp−1X (i− d))


Proof. Corollary 16.1.3 yields a recurrence

χ(ΩpP(i)) =(n+ 1p

)χ(OP(i− p))− χ(Ωp−1

P (i))

Repeated application of this proves (a).Lemma 16.2.3 and corollary 16.2.5 implies

χ(ΩpX(i)) = χ(ΩpP(i)|X)− χ(Ωp−1X (i− d))

= χ(ΩpP(i))− χ(ΩpP(i− d))− χ(Ωp−1X (i− d))

When p = 0, the right side can be evaluated explicitly to obtain (b).

Corollary 16.3.3. The Hodge numbers of X depend only on d and n, andare given by polynomials in these variables. Specifically

hn0 = (−1)n(n− dn

)=(d− 1n

)We give a generating function below. Let hpq(d) denote the pqth Hodge number

of smooth hypersurface of degree d in Pp+q+1. Define the formal power series

H(d) =∑pq

(hpq(d)− δpq)xpyq

in x and y. Then:

Theorem 16.3.4 (Hirzebruch).

H(d) =(1 + y)d−1 − (1 + x)d−1

(1 + x)dy − (1 + y)dx

Corollary 16.3.5. If X ⊂ Pn+1 has degree 2, then bn(X) = 0 if n is odd,otherwise bn(X) = hn/2,n/2(X) = 2.

Proof.

H(2) =1

1− xy

By expanding the series H(3) for a few terms, we can see that

Corollary 16.3.6. If X ⊂ Pn+1 has degree 3, the middle hodge numbers are

1, 1

0, 7, 0

0, 5, 5, 0

0, 1, 21, 1, 0

0, 0, 21, 21, 0, 0

for n ≤ 5

16.4. DOUBLE COVERS 211

Hirzebruch [Hrz, 22.1.1] deduced a slightly different identity from his generalRiemann-Roch theorem. The above form of the identity appears in [SGA7, expXI cor. 2.4]. Similar formulas are available for complete intersections. This resultcan be deduced from the previous formulas in principle, but this could get quitemessy. We will be content to work out the case where y = 0. On one side we havewe have generating function

∑hn0(d)xn By corollary 16.3.3, this equals∑(d− 1n+ 1

)xn =

(1 + x)d−1 − 1x

which is what one gets by substituting y = 0 into Hirzebruch’s formula.

Exercises

1. Calculate the Hodge numbers of a degree d surface in P3 using the recur-rence formulas.

2. Prove that for every fixed d and p, there exists q0 such hpq(d) = 0 forq ≥ q0.

16.4. Double covers

Our goal is to compute the Hodge numbers for another natural class of ex-amples which generalize hyperelliptic curves. Let f(x0, . . . xn) ∈ C[x0, . . . xn] bea homogeneous polynomial of degree 2d such that the hypersurface D ⊂ Pn = Pdefined by f = 0 is nonsingular. Let π : X → P be the double cover branchedover D (example 3.5.1). By construction this is gotten by gluing the affine varietiesdefined by y2 = f(1, x1, . . . xn), . . .. It follows that X is nonsingular.

Lemma 16.4.1 (Esnault-Viehweg).

π∗ΩpX∼= ΩpP ⊕ (ΩpP(logD)⊗O(−d))

Sketch. The Galois group, which is generated by the involution σ : y 7→ −y,acts on π∗Ω

pX . So we can decompose this into a direct sum of the σ-invariant part,

and the part where σ acts by −1. We check that these summands correspond to ΩpPand ΩpP(logD)⊗O(−d) respectively. It is enough to do this for the analytic sheaves.By the implicit function theorem, we can choose new analytic local coordinates suchthat X is given locally by y2 = x1. A local basis for π∗Ω1

X is given by

ydy =12dx1, dx2, . . . dxn; dy =

√x1dx1

2x1, ydx2, . . . , ydxn

The first n forms are invariant and form a local basis for Ω1P. The remainder are

anti-invariant and form a local basis for Ω1P(logD)⊗O(−d). By taking wedges, we

get a similar decomposition for p-forms. See [EV, pp 6-7] for further details.

Corollary 16.4.2.

Hq(X,ΩpX) ∼= Hq(P,ΩpP)⊕Hq(P,ΩpP(logD)⊗O(−d))

Proof. Let Ui be the standard affine cover of Pn. Then Ui = π−1Ui givesan affine cover of X. We can compute Hq(X,ΩpX) using the Cech complex

C(Ui,ΩpX) = C(Ui,ΩpP)⊕ C(Ui,ΩpP(logD))

to obtain the sum of the above spaces.


Corollary 16.4.3. We have

hpq(X) = δpq + dim coker[Hq−1(Ωp−1D (−d))→ Hq(ΩpP(−d))]

+ dim ker[Hq(ΩpD(−d))→ Hq+1(ΩpP(−d))]

in general, and hpq(X) = δpq if p+ q < n.

Proof. This follows from (11.6.3) from § 11.6 together with Bott’s vanishingtheorem 16.1.5.

We can obtain more explicit formulas by combining this with the earlier results.

Exercises

1. When n = 2, check that h02(X) = (d−1)(d−2)2 and h11(X) = 3d2 − 3d+ 2.

2. Verify that hpq(X) = δpq also holds when p+ q > n.

16.5. Griffiths Residues

In this section we describe an alternative method for computing the Hodgenumbers of a hypersurface due to Griffiths. Although the point is really that thisgives more, namely a method for computing the Hodge structure (or more preciselythe part one gets by ignoring the lattice). Further details can be found in [CMP,§3.2] and [V, vol II, chap 6]. We work over C in this section.

Suppose that X ⊂ Y = P = Pn+1 is a smooth hypersurface defined by apolynomial f ∈ C[x0, . . . xn+1] of degree d. Then Hn−1(P) ∼= Hn−1(X). ThereforeHn+1(U) maps isomorphically onto the primitive cohomology Pn(X) = ker[Hn(X)→Hn+2(P)]. This is the same as Hn(X) if n is odd, and has dimension one less if nis even. The Hodge filtration on F pHn+1(U) maps onto the Hodge filtration on Xwith a shift F p−1Pn(X). The next step is show that the Hodge filtration agreeswith the pole filtration in this case. Then in summary:

Theorem 16.5.1 (Griffiths). The Hodge filtration F pPn(X) coincides with thepole filtration on Hn+1(U) with a shift by +1. This can be identified with thequotient

H0(Ωn+1P ((n− p+ 1)X))

dH0(ΩnP((n− p)X))

Proof. For p = n, this is immediate

FnPn(X) = Fn+1Hn+1(U) = Hn+1(Ωn+1P (logX)) = Hn+1Ωn+1

P (X))

For p = n− 1, we use the exact sequence

0→ ΩnP,cl(logX)→ ΩnP(X) d→ Ωn+1P (2X)→ 0

where ΩP,cl(. . .) is the subsheaf of closed forms. Then

H0(ΩnP(X)) d→ H0(Ωn+1P (2X))→ H1(ΩnP,cl(logX))→ H1(ΩnP(X))

On the right, the group H1(ΩnP(d)) = 0 by theorem 16.1.5. Thus FnHn+1(U) =H1(ΩnP,cl(logX)) is described as in the theorem. The remaining p’s can be handledby a similar argument.

This can be made explicit using:

16.5. GRIFFITHS RESIDUES 213

Lemma 16.5.2. Let

ω =∑

(−1)ixidx0 ∧ . . . dxi . . . dxn+1

Then

H0(Ωn+1P (kX)) =

gω

fk| g homogeneous of deg = kd− (n+ 2)

With these identifications, elements of Pn(X) can be represented by homoge-

neous rational differential forms gω/fk modulo exact forms. Set

R =C[x0, . . . xn+1]

(∂f/∂x1, . . . , ∂f/∂xn+1)The ring has a grading R = ⊕Ri coming from the polynomial ring. We define amap Pn(X)→ R by sending the class of gω/fk to the class of g.

Theorem 16.5.3 (Griffiths). Under this map, the intersection of Hn−p(X,ΩpX)with Pn(X) maps isomorphically to Rτ(p), where τ(p) = (n− p+ 1)d− (n+ 2).

This leads to an alternative method for computing the Hodge numbers of adegree d hypersurface.

Corollary 16.5.4. hpq(d)−δpq is the coefficient of tτ(p) in (1+t+. . .+td−2)n+2

Proof. We can assume that f = xd0 + xd1 + . . . is a Fermat equation. Itsuffices to prove that the Poincare series of R, which is the generating functionp(t) =

∑dimRit

i, is given by

p(t) = (1 + t+ . . . td−2)n+2

Note that

R =C[x0, . . . xn+1]

(xd−10 , xd−1

1 , . . .)∼=

C[x](xd−1)

⊗ C[x](xd−1)

. . . (n+ 2 times)

Since Poincare series for graded rings are multiplicative for tensor products, p(t) isthe (n+ 2) power of the Poincare series of C[x]/(xd−1), and this is given by aboveformula.

Exercises

1. Using exact sequences

0→ Ωn−iP,cl ((j − 1)X)→ Ωn−iP ((j − 1)X)→ Ωn−i+1P (jX)→ 0

and identifications

F pHn+1(U) ∼= Hn+1−p(ΩpP,cl(logX)) ∼= Hn+1−p(ΩpP,cl(X))

finish the proof of theorem 16.5.1.

CHAPTER 17

Deformation invariance of Hodge numbers

In the previous chapter, we observed that the Hodge numbers of a hypersurfacein projective space depend only the degree. This is refined by a theorem of Kodairaand Spencer that the Hodge numbers of a family of smooth complex projectivevarieties stays constant. Our goal is to prove this. While the original proof wasentirely analytic, we can appeal to the GAGA theorem 15.5.2 to do much of thegroundwork in the algebraic setting. We do this since Grothendieck’s language ofschemes gives a particularly elegant approach to families.

17.1. Families of varieties via schemes

Varieties often occur in families as we have already seen. As a simple example,y2 = xt defines a family of parabolas in the xy-plane A2

k parameterized by t ∈ A1k.

When t = 0, we get a degenerate parabola y2 = 0 which is the “doubled” x-axis.It is impossible to capture this fully within the category of varieties, but it makesperfect sense with schemes (see §3.4). Here we have a map of rings

k[t]→ k[x, y, t](y2 − xt)

which induces a morphism of schemes

π : Mspeck[x, y, t](y2 − xt)

→ Mspec k[t] = A1k

The fibres of π gives a family of schemesMspec

k[x, y](y2 − at)

⊂ A2k | a ∈ k

.

More generally, let R = O(Y ) be the coordinate ring be an affine variety Yover an algebraically closed field k. Affine space over R is

AnR = MspecR[x1, . . . xn] ∼= Ank × Y.Let I ⊂ R[x1, . . . xn] be an ideal. Choose generators

fj(x, y) =∑

fj,i1,...in(y)xi11 . . . xinn

for I. We get a morphism

MspecR[x1, . . . xn]/I → MspecR = Y

We can view this as a family of closed subschemes of AnkMspecR[x1, . . . xn]/I ⊗R/ma = Mspec k[x1, . . . xn]/(fj(x, a))

parameterized by points a ∈ Y .Similarly, a homogenous ideal I ⊂ R[x1, . . . xn] = S gives a family Proj S/I

of subschemes of Pnk . A precise definition of Proj can be found in [Har]. We will

215

216 17. DEFORMATION INVARIANCE OF HODGE NUMBERS

describe this in terms that are most convenient for us. If I has homogenous gener-ators fj(x, y). We can loosely define Proj S/I as the family of subschemes definedby ideals (fj(x, a)) as a ∈ Y varies. A bit more formally, given a finitely generatedgraded R-module M , we can construct a coherent sheaf M on projective space overY : PnY = PnR = Pnk×Y as in section 15.3. Theorem 15.3.4 generalizes to this setting[Har, II, ex. 5.9]. For each point a ∈ Y , we get a graded k[x0, . . . xn]-moduleMa = M ⊗R/ma. Thus we get a family of coherent sheaves Ma parameterized byY . Alternatively, let ia : Pnk → Pnk×Y be the inclusion x 7→ (x, a), then Ma = i∗aM .We can apply this construction to get a sheaf of ideals I on PnR, Proj S/I is thecorresponding closed subscheme. This fits into a diagram

Proj S/I

$$HHHHHHHHH// PnR = Pnk × Y

yysssssssssss

Y

We look at various properties of families. Smoothness is the algebraic ana-logue of a submersion. To simply matters, we restrict our attention to nonsingularvarieties over an algebraically closed field.

Definition 17.1.1. A morphism f : X → Y of nonsingular varieties is smoothif for every point x, the induced map Tx → Ty is surjective.

For a morphism of general schemes this definition is too naive. See [Har, III§10]for the correct definition and further discussion.

Example 17.1.2. Let f : Amk → Ank be given by f(x) = (f1(x), . . . fn(x)). wherefi are polynomials. Then f is smooth if and only if the Jacobian (∂fi/∂xj) has rankn. When this is satisfied, it is clear that the fibres f−1(y) are nonsingular varietiesof dimension m− n.

In general, if k is algebraically closed then:

Lemma 17.1.3. The (closed) fibres f−1(y) of a smooth morphism of nonsingularvarieties are nonsingular.

Thus we can view a smooth morphism as a family of nonsingular varieties overa nonsingular parameter space. A more flexible notion is flatness.

Definition 17.1.4. An R-module T is flat if the functor N 7→ T ⊗R N is leftexact and therefore exact. If Y = MspecR a morphism f : X → Y of schemes isflat if O(U) a flat R-module for every open set U ⊆ X.

The geometric meaning of flatness is a bit elusive, and we will try to discover themeaning shortly. For the moment, the quickest explanation is provided by Mumford[Mu2], “flatness is a riddle that comes out of algebra, but which is technically theanswer to many prayers.” Basic examples are provided by:

Lemma 17.1.5. A smooth morphism of nonsingular varieties is flat.

Proof. Flatness is usually taken as part of the definition. However, a proofusing our definition can be found in [Har, III prop 10.4].

Exercises

17.2. SEMICONTINUITY OF COHERENT COHOMOLOGY 217

1. Prove the following examples are flat R-modules:(a) Locally free modules.(b) Unions of a directed families of locally free modules. A family of

subsets is directed, or filtered, if the union of any pair of members ofthe family is contained in another member.

(c) R[x1, . . . xn].(d) R[[x1, . . . xn]].

2. Assume the following: A finitely generated module M is a flat R if andonly each localization My is flat over Ry if and only if each completionMy is flat over Ry, for y ∈ Y . Then prove lemma 17.1.5. (Hint: Provethe map on completed local rings is given by Oy → Oy[[x1, . . . xr]] wherer = dimX − dimY .)

3. Show that the blow up morphism

(x, `) ∈ A2 × P2 | x ∈ ` → A2

is not flat.

17.2. Semicontinuity of coherent cohomology

Let R = O(Y ) be the coordinate ring of an affine variety, and let

S = R[x0, . . . xn] =⊕

Si

with the usual grading. Given a finitely generated graded R-module M , we definedthe coherent sheaf M on PnR = Pnk × Y earlier. When applied to a homogeneousideal I ⊂ S, this gives rise to a family of projective schemes of Pnk with ideal sheavesIy. Another basic example is:

Example 17.2.1. Let M be a finitely generated graded k[x0, . . . xn]-module,then R⊗M is a finitely generated graded S-module. R⊗M is a “constant” familyof sheaves; we have R⊗My = M . This can be constructed geometrically. Letπ : Pnk × Y → Y be the projection. Then R⊗M = π∗(M). A special case isOPn

k×Y (i) which is the constant family π∗OPnk

.

The basic question is: given a graded module M as above, how do the dimen-sions of the cohomology groups

hi(My) = dimHi(My)

vary with y ∈ Y ? Consider some examples.

Example 17.2.2. If M is constant family, then clearly y 7→ hi(My) is a con-stant function of y.

In general, this is not constant. Here is a more typical example:

Example 17.2.3. Set R = k[s, t], and choose three points

p1 = [0, 0, 1], p2 = [s, 0, 1], p3 = [0, t, 1]

in P2k with s, t variable. Let I be the ideal sheaf of the union of these points in

P2k × A2

k. This can also be described as I, where I is product

(x, y, z − 1)(x− s, y, z − 1)(x, y − t, z − 1) ⊂ R[x, y, z]


Consider I(1) = I(1). The global sections of this sheaf correspond to the space oflinear forms vanishing at p1, p2, p3. Such a form can exist only when the points arecolinear, and it is unique upto scalars unless the points coincide. Thus

h0(I(1)(s,t)) =

2 if s = t = 01 if s = 0 or t = 0 but not both0 if s 6= 0 and t 6= 0

In the above example, the sets where the zeroth cohomology jumps are Zariskiclosed. More generally:

Theorem 17.2.4 (Grothendieck). Let R be a finitely generated algebra, andY = MspecR. Suppose that M is a finitely generated graded S = R[x0, . . . xn]-module which is flat as an R-module. Then

y 7→ hi(My)

is upper semicontinuous i.e. the sets y | hi(My) ≥ r are Zariski closed. TheEuler characteristic

y 7→ χ(My) =∑

(−1)ihi(My)

is locally constant.

We postpone the proof until the next section. Grauert has established analo-gous results in the analytic setting, see [GPR]. The following corollary, which is infact an if and only if [Har, III, 9.9], explains the significance of flatness. It impliesthat maps, such as blow ups, where the fibre dimension jump cannot be flat.

Corollary 17.2.5. If X → Y is flat family of projective schemes, then theHilbert polynomial χ(OXy

(d)) is locally constant. In particular, the fibres have thesame dimension over connected components.

We can define an equivalence relation on nonsingular projective varieties weakerthan isomorphism. We say that two nonsingular projective varieties X1, X2 aredeformations of each other if there is a smooth map of nonsingular varieties f :Z → Y such that the fibres are all projective varieties and Xi = f−1(yi) forpoints yi ∈ Y . This generates an equivalence relation that we will call deformationequivalence. For example, any two nonsingular hypersurfaces Xi ⊂ Pn of degreed are deformation equivalent, since they members of the the family constructedbelow. Let Vd be the space of homogeneous polynomials of degree d in n + 1variables. ∆ ⊂ P(Vd) the closed set of singular hypersurfaces. Then

U = (x, [f ]) ∈ Pn × (P(Vd)−∆) | f(x) = 0Then U → P(Vd)−∆ is a smooth map containing all nonsingular hypersurfaces asfibres.

Since any elliptic curve can be realized as a smooth cubic in P2, it follows thatany two elliptic curves are deformation equivalent. Considerably deeper is the factthat any two smooth projective curves of the same genus g ≥ 2 are deformationequivalent. This can be proved with the help of the Hilbert scheme. A weakformulation of its defining property is as follows:

Theorem 17.2.6 (Grothendieck). Fix a polynomial p ∈ Q[t]. There is a pro-jective scheme H = HilbpPn with a flat family U → H of closed subschemes of Pnwith Hilbert polynomial p, such that every closed subscheme with Hilbert polynomialp occurs as a fibre of U exactly once.

17.3. PROOF OF THE SEMICONTINUITY THEOREM 219

For example, the space P(Vd) above is the Hilbert scheme HilbpPn with

p(i) = χ(OPn(i))− χ(OPn(i− d)) =(n+ i

n

)−(n+ i− d

n

).

Choose N ≥ 3, then for a smooth projective curve X of genus g, ω⊗NX is very ample.The set of such curves in P(g−1)(2N−1) is parameterized by an open subset of theHilbert scheme with p(t) = (2g − 2)Nt + (1 − g). This set can be shown to beirreducible. See [HM] for further details.

Exercises

1. Give an explicit example where h1 jumps.2. The arithmetic genus of a possibly singular curve X is 1−χ(OX). Calculate

it for a plane curve.

17.3. Proof of the semicontinuity theorem

Everything in this section works for the case of R a noetherian ring, andY = SpecR. But for simplicity, let R be a finitely generated k-algebra withY = MspecR.

Definition 17.3.1. An R-module I is injective if and only if for any for injec-tive map N → M of R-modules, the induced map HomR(M, I) → HomR(N, I) issurjective.

Example 17.3.2. This notion makes sense for any ring. If R = Z then I isinjective provided that it is divisible i.e. if a = nb has a solution for every a ∈ Iand n ∈ Z− 0.

A standard result of algebra is the following.

Theorem 17.3.3 (Baer). Every module is (isomorphic to) a submodule of aninjective module.

Proof. A proof can be found for example in [E, p. 627].

Given an R-module, we have the associated sheaf M on Y , (section 3.6).

Proposition 17.3.4. If I is an injective R-module, then I is flasque.

Proof. A complete proof can be found in [Har, III, 3.4]. We sketch this underthe extra assumption that R is an integral domain i.e. when Y is a variety. Supposethat U = D(f) is a basic open set. Given an element of I(U), it can be written as afraction x/fn. Consider the map ξ : R→ I which sends 1 7→ x. Since I is injectiveand R → R[1/f ] is one to one by assumption, ξ extends to a map R[1/f ] → I.Therefore x/fn = ξ(1/fn) lies in I.

By induction on dimension of support (or more generally noetherian induction),we can assume that the proposition holds for any injective module whose associatedsheaf is supported on a proper subset. For a general open set U , choose a nonemptyD(f) ⊆ U , and let Z = Y − D(f). Let J ⊂ I be the submodule consisting ofelements annihilated by a power of f . The key point is that J is also injective[Har, III, 3.2] and J is supported in Z. So that J is flasque by induction. Onechecks that

J(U)→ I(U)→ I(D(f))


is exact. Since the sections at both ends extend to Y , the same goes for I(U).

Theorem 17.3.5 (Grothendieck). Let Y = MspecR, then for any R-moduleM ,

Hi(Y, M) = 0

Proof. The module M embeds into an injective module I. Let N = I/M . Bylemma 3.6.11, M 7→ M is exact, therefore

0→ M → I → N → 0

is exact. As Hi(Y, I) = 0 for i > 0, we obtain

I → N → H1(Y, M)→ 0,

andHi+1(Y, M) ∼= Hi(Y, N).

Since I → N is surjective, H1(Y, M) = 0. Since this argument can be applied toany module, and in particular to N , we see that

H2(Y, M) ∼= H1(Y, N) = 0.

We can kill the all higher cohomology groups in the same fashion.

Corollary 17.3.6. An affine cover of a variety is Leray with respect to anycoherent sheaf.

Let us return to the setup of the previous section. Let R be the coordinatering of an affine variety Y defined over k. Let S = R[x0, . . . xn], and suppose thatM is a graded a finitely generated S-module. Then we can form the coherent sheafM on PnR as above. Let C•(M) denote the Cech complex of M with respect to thestandard affine cover of PnR. This is a Leray cover by the theorem, therefore:

Corollary 17.3.7. Hi(C•(M)) ∼= Hi(PnR, M).

Most of the arguments of chapter 15 can be reprised to prove that these coho-mologies are finitely generated as R-modules [Har]. Of course, we really want thecohomology of the fibre Hi(Pnk , My), which is computed using the Cech complexC•(My). When M is flat, there is a simple relationship with the previous com-plex, namely C•(M)⊗R/my

∼= C•(My). Thus C•(M)⊗R/my computes Hi(My).C•(M) can be replaced by a complex D• of finitely generated locally free moduleswith the same property by:

Lemma 17.3.8. Suppose that C• is a finite complex of flat R-modules withfinitely generated cohomology. Then there exists a finite complex D• of finitelygenerated locally free modules, and map of complexes D• → C•, such that

Hi(D• ⊗N) ∼= Hi(C• ⊗N)

for any R-module N .

Proof. [Mu1, pp. 47-49].

Proof of theorem 17.2.4. We assume that a complex D• of finitely gener-ated locally free modules has been constructed as above. By shrinking Y if neces-sary, there is no loss in assuming that these are free modules. Thus the differentials

17.4. DEFORMATION INVARIANCE OF HODGE NUMBERS 221

Di → Di+1 can be represented by matrices Ai over R = O(Y ). The differentialsA•(y) of D• ⊗R/my are obtained by evaluating these matrices at y. We note that

hi(My) = rk(Di)− rk(Ai(y))− rk(Ai−1(y))

Thus the inequality hi(My) ≥ r can be expressed in terms of vanishing of theappropriate minors of these matrices, and this defines a Zariski closed set. Bylemma 11.4.6, χ(My) is the alternating sum of ranks of the Di. This is independentof y.

Exercises

1. Show that theorem 17.3.5 fails for j!OU as defined in exercises of §15.2.2. Show that Riπ∗(M) ∼= Hi(D•) with the above notation.3. Show that the functions hi(My) are all locally constant if and only if the

sheaves Riπ∗(M) are all locally free, where π : PnR → Y is the projection.

17.4. Deformation invariance of Hodge numbers

In this section, we revert to working over C.

Theorem 17.4.1 (Kodaira-Spencer). If two complex nonsingular projective va-rieties are deformation equivalent, then their Hodge numbers are the same.

Proof. Let f : X → Y be a smooth projective morphism of nonsingularvarieties, with Y connected. Then by theorem 17.2.4, there there are constants gpq

such that

(17.4.1) hpq(Xt) ≥ gqp

for all t ∈ X with equality on an a nonempty open set U . Choose a points t ∈ Uand s /∈ U . Then since Xt and Xs are diffeomorphic 13.1.1, they have the sameBetti numbers. Therefore by the Hodge decomposition (theorem 10.2.4),∑

pq

hpq(Xs) =∑pq

gpq

and this implies that (17.4.1) is an equality.

This result relies on the Hodge theorem in an essential way, and is known tofail when C is replaced by a field of positive characteristic. Also it can fail formany of the other invariants considered so far, even over C. Any elliptic curvecan be embedded as a cubic in P2, thus any two elliptic curves are deformationequivalent. Likewise for the products of elliptic curves with themselves. But wesaw in section 12.2 that the Picard number of E × E was not constant. Thereforeit not a deformation invariant. Other examples of this phenomenon are providedby the next theorem and the exercises.

Theorem 17.4.2 (Noether-Lefschetz). Let d ≥ 4 then there exists a surfaceX ⊂ P3 of degree d with Picard number 1.

Remark 17.4.3. Noether refers not to Emmy, but her father Max who was alsoa mathematician.

Remark 17.4.4. The result is true “for almost all” surfaces. See the exercises.


Sketch. Since H1(X,OX) = 0 for any surface in P3, c1 : Pic(X)→ H2(X,Z)is injective. On the other hand, H2(X,OX) 6= 0 if X is has degree d ≥ 4, thereforec1 is not onto. Choose a Lefschetz pencil Xtt∈P1 of surfaces, let X → P1 bethe incidence variety and let U ⊂ P1 parameterize Xt smooth (see 14.4). The setof all curves lying on some Xt is a parameterized by a countable union of Hilbertschemes. Each irreducible component will either parameterize curves lying on afixed Xt or curves varying over all Xt. Therefore for all but countably many t, anycurve lying on a an Xt will propogate to all the members of the pencil. Choosesuch a nonexceptional t0 ∈ U . The group c1(Pic(Xt0)) is the group of curves onXt0 , and this is stable under the action of π1(U, t0) since any such curve can bemoved along a loop avoiding the exceptional t’s. By theorem 14.4.9, H2(Xt0 ,Q) =im(H2(P3)) ⊕ V , where V is the generated by vanishing cycles, and π1(U) actsirreducibly on this. Since C = c1(Pic(Xt0)) ⊗ Q contains im(H2(P3)), it followsthat either c1(Pic(Xt0)) equals im(H2(P3)) or it equals H2(Xt0). The last case isimpossible, so c1(Pic(Xt0)) = im(H2(P3)) = Q.

Exercises

1. Is the restriction d ≥ 4 in Noether-Lefschetz necessary?2. Let X ∈ P3 be a smooth quartic containing a line ` (refer back to the

exercises of § 12.1). Prove that the Picard number of X is at least 2.3. Using the Baire category theorem (that a countable union of nowhere dense

sets in a complete metric space is nowhere dense), show that the subset ofsurfaces for which the conclusion of Noether-Lefschetz fails is nowhere densein the projective space of all surfaces of degree d ≥ 4.

Part 5

A Glimpse Beyond*

CHAPTER 18

Analogies and Conjectures

In this final chapter, we end our story by beginning another. Although ithas been clear from our presentation that we can do algebraic geometry over anyfield. Each field has its own character: transcendental over C, and arithmetic overfields such as Q,Fp . . .. It may seem that the arithmetic and transcendental sideswould have very little to do with each other, but in fact they are related in deep andmysterious ways. We start by briefly summarizing the results of Weil, Grothendieckand Deligne for finite fields. Then we return to complex geometry and prove Serre’sanalogue of the Weil conjecture. This result inspired Grothendieck to formulate hisstandard conjectures. We explain some of these along with the closely related Hodgeconjecture. These are among the deepest open problems in algebraic geometry.

18.1. Counting points and Euler characteristics

Let Fq be the field with q = pr elements, where p is a prime number. Thealgebraic closure k = Fp = ∪Fqn . Suppose that X ⊆ Pdk is quasiprojective varietydefined over Fp, that is assume the coefficients of the defining equations lie in Fq.Let X(Fqn) to be the set of points of PNFqn

satisfying the equations defining X. LetNn(X) be the number of points of X(Fqn). Here are a few simple computations:

Example 18.1.1. Nn(AmFp) = pnm.

Example 18.1.2. Expressing Pm = Am ∪Am−1 ∪ . . . as a disjoint union yields

Nn(PmFp) = pnm + pn(m−1) + . . . pn + 1.

Example 18.1.3. Nn((P1Fp

)m) = (pn + 1)m

The last two computation is based on the following obvious properties. Nn isadditivity:

Nn(X) = Nn(X − Z) +Nn(Z)whenever Z ⊂ X is closed. Nn is multiplicative:

Nn(X × Y ) = Nn(X)Nn(Y )

Now let us return to complex geometry so that k = C. The Euler characteristicwith respect to compactly supported cohomology is

χc(X) =∑

(−1)i dimHic(X,R)

It is not difficult to compute this number in the above examples using the techniquesfrom the earlier chapters.

χc(AmC ) = 1χc(PmC ) = m

χc((PC)m) = 2m

225

226 18. ANALOGIES AND CONJECTURES

This leads to the following curious observation that if we set p = 1 in the aboveformulas then we get χc. In some sense there is nothing to prove, but we can atleast give a more satisfying explanation. First note that although defined compactlysupported cohomology using differential forms in section 5.3, there is a purely topo-logical definition which works for any locally compact Hausdorff space. We can set

Hic(X) = Hi(X, X −X)

for any (or some) compactification X. Then (7.2.1) and a little diagram chasingyields the long exact sequence

(18.1.1) . . . Hic(X)→ Hi

c(X)→ Hic(X − Z)→ Hi+1

c (X)→ . . .

The first clue that there is a deeper relation between Nn and χc is the following.

Lemma 18.1.4. Let χc is an additive and multiplicative invariant, i.e.

χc(X) = χc(X − Z) + χc(Z)

χc(X × Y ) = χc(X)χc(Y )

holds.

Proof. The additivity follows immediately from (18.1.1). The multiplicativityfollows from the Kunneth formula

Hic(X × Y,R) =

⊕j+l=i

Hjc (X,R)⊗H l

c(Y,R)

Suppose, we start with a complex quasiprojective algebraic variety X with afixed embedding into PNC . If the defining equations (and inequalities) have integercoefficients, then we can reduce these modulo a prime p to get a quasiprojectivevariety (or more accurately scheme) Xp defined over the finite field Fp. In lessprosaic terms, we have a scheme over Spec Z, and Xp is the fibre over p. (Inpractice, Z might be replaced by something bigger such as the ring of integers of anumber field, but the essential ideas are the same.) Then we can count points onXp and compare it to χc(X). To avoid certain pathologies, we should take p largeenough.

Lemma 18.1.5. If X is expressible as a disjoint union of affine spaces thenNn(Xp) is a polynomial in pn for sufficiently large p. Substituting p = 1 yieldsχc(X).

Proof. If X =∐

Ami , then we get a similar decomposition for Xp with p 0.Therefore Nn(Xp) =

∑pnmi and χc(X) =

∑1nmi .

The lemma applies to the above examples of course, as well as to the largerclass of toric varieties [F2, p 103], Grassmanians and more generally flag varieties[F1, 19.1.11]. Nevertheless the most varieties do admit such decomposition.

18.2. THE WEIL CONJECTURES 227

18.2. The Weil conjectures

One could ask whether something like lemma 18.1.5 holds for arbitrary varieties.We can start by looking at an elliptic curve which is the simplest example wherethe lemma does not apply.

Example 18.2.1. Let us take the curve X given by the affine equation y2 =x3 − 1. This defines a nonsingular curve Xp over Fp when p ≥ 5. So let usanalyze p = 5. When n is odd, 5n − 1 is not divisible by 3. This implies thatx 7→ x3 is an automorphism of F5n . Therefore y2 + 1 has a unique cube root. ThusNn(X5) = 1 + 5n if n is odd. When n is even, we can compute a few values bybrute force on a machine.

pn Nn52 36 = 1 + 52 + 2 · 554 576 = 1 + 54 − 2 · 52

56 15876 = 1 + 56 + 2 · 53

58 389376 = 1 + 58 − 2 · 54

510 9771876 = 1 + 510 + 2 · 55

So at least empirically, we have a formula in the previous example

Nn(X5) =

1 + 5n if n odd1 + 5n − 2(−5)n/2 if n even

= 1 + 5n − (√−5)n − (−

√−5)n

Example 18.2.2. Calculating the number of points for y2 = x3−x with p = 3,we get

pn Nn3 4 = 1 + 332 16 = 1 + 33 + 2 · 333 28 = 1 + 33

34 64 = 1 + 33 − 2 · 32

35 244 = 1 + 35

ThenNn(X3) = 1 + 3n − (

√−3)n − (−

√−3)n

fits this data.

At this point, one might guess Nn is always 1 + pn − (√p)n − (−√p)n for an

elliptic curve, but it is a bit more complicated (see exercises). We get cheap higherdimensional examples, by taking products of these. Based on these examples, onemight guess that in general: Nn(Xp) is a polynomial in expressions λni and settingλi = 1 yields the Euler characteristic. This turns out to be correct, but it seemsto come out of nowhere when stated like this. We need some guiding principle tounderstand these formulas better. The basic insight goes back to Weil. Supposethat X ⊂ Pd is a nonsingular projective variety with equations defined over Z asabove. Let us denote by Xp the variety over the algebraic closure Fp determinedby reducing the equations modulo p. The Frobenius morphism F : Xp → Xp

is the map which raises the coordinates to the pth power (see [Har] for a moreprecise description). Then Xp(Fpn) are the points of Xp fixed by Fn. If this were amanifold with a self map F (satisfying appropriate transversallity conditions), then


we could calculate this number using the Lefschetz trace formula. Weil conjecturedthat such a cohomology theory should exist in the present setting. Grothendieckeventually constructed such a Weil cohomology theory, in fact several. For eachprime ` 6= p, he constructed functors Hi

et(−,Q`) called `-adic cohomology suchthat:

1. Hiet(Xp,Q`) is a vector space over the field of `-adic numbers Q` = (lim←−Z/`n)⊗

Q.2. dimHi

et(Xp,Q`) is the usual ith Betti number of X.3. F acts on these spaces. The action on H0

et(Xp,Q`) is trivial, and it acts bymultiplication by pdimX on H2 dimX

et (Xp,Q`).4. There is a Lefschetz trace formula which implies that

Nn(Xp) =∑i

(−1)itrace[Fn∗ : Hiet(Xp,Q`)→ Hi

et(Xp,Q`)]

Details can be found in [Mi]. The last formula can be rewritten as

Nn(Xp) =∑i

(−1)i∑

λnij

where λij are the generalized eigenvalues of F ∗. In the previous examples, thenumbers ±

√−5,±

√−3 above were precisely the eigenvalues of F acting on H1.

Although this is overkill, we can also use this formalism to reprove the formulas ofthe last section. For example,

Hiet(Pm,Q`) =

Q` with F acting by pi/2 if i < 2m is even0 otherwise

gives the formula for Nn(Pm). Notice that the absolute value of the eigenvaluesin these examples have very specific sizes. This is consistent with a deep theoremof Deligne proving the last of Weil’s conjectures. (For more background and inparticular what this has to do with the Riemann hypothesis, see [Har, appendixC], [K], [Mi]).

Theorem 18.2.3 (Deligne). Let X be a smooth and projective variety definedover Fq, and X its extension to Fq. Then the eigenvalues of the qth power FrobeniusHiet(X,Q`) are algebraic numbers λ all of whose absolute values satisfy |λ| = qi/2.

Eigenvalue above really means generalized eigenvalue. The action of F hasbeen conjectured to be diagonalizable, this is still open. This abstract theorem hasconcrete consequences.

Corollary 18.2.4. Let X ⊂ PN+1 be a smooth degree d hypersurface definedover Fp then

|Nn(X)− (1 + p+ . . . pN )| ≤ bN · pN/2

where bN is the N th Betti number of a smooth degree d hypersurface in PnC.

Proof. We can assume that X comes from a hypersurface over C by reducingmodulo a prime. By results proved earlierHi(X,Q) ∼= Hi(PN+1,Q) for i ∈ [0, 2N ]−N. So the Betti numbers of X and PN+1 are the same in this range. In fact,the action of F would compatible with this isomorphism, eigenvalues would be the

18.3. A TRANSCENDENTAL ANALOGUE OF WEIL’S CONJECTURE 229

same of both spaces for i ∈ [0, 2N ] − N. For PN+1, these are given above. LetλjN denote the eigenvalues on the Nth cohomology of X. Then

|N1(X)− (1 + p+ . . . pN )| ≤ |bN∑j=1

λjN | ≤ bN · pN/2

When X is singular or open, then the above theorem is no longer true. Delignehas shown that the eigenvalues can have varying sizes or weights. If one countsthe number of eigenvalues on Hi(Xp) of a given weight, then this also has a tran-cendental interpretation in terms of the mixed Hodge structure on Hi(X). See[D3].

Exercises

1. Find λ for which Nn = 1+pn−λn−λn fits the data for y2 = x3−x+1, p = 3.pn Nn3 732 733 2834 9135 217

18.3. A transcendental analogue of Weil’s conjecture

After this excursion into arithmetic, let us return to Hodge theory and provean analogue of the Weil’s Riemann hypothesis found by Serre [S3]. To set up theanalogy let us replace X above by a smooth complex projective variety Y , andF by and endomorphism f : Y → Y . As for q, if we consider the effect of theFrobenius on PNFq

, the pullback of O(1) under this map is O(q). To complete theanalogy, we require the existence of a very ample line bundle OY (1) on Y , so thatf∗OY (1) ∼= OY (1)⊗q. We can take c1(OY (1)) to be the Kahler class ω. Then wehave f∗ω = qω.

Theorem 18.3.1 (Serre). If f : Y → Y is holomomorphic endomorphism of acompact Kahler manifold with Kahler class ω, such that f∗ω = qω for some q ∈ R.Then q is algebraic integer, f∗ : Hi(Y,Q) → Hi(Y,Q) is diagonalizable and itseigenvalues are algebraic integers with absolute value qi/2.

Proof. The theorem holds for H2n(Y ) since ωn generates it. Note qn is thedegree of f which is necessarily a (rational) integer. Therefore q is an algebraicinteger. By hypothesis, f∗ preserves the Lefschetz decomposition (theorem 14.1.1),thus we can replace Hi(Y ) by primitive cohomology P i(Y ). Recall from corol-lary 14.1.4,

Q(α, β) = Q(α,Cβ)

is positive definite Hermitean form P i(Y ), where

Q(α, β) = (−1)i(i−1)/2

∫α ∧ β ∧ ωn−i.


Consider the endomorphism F = q−i/2f∗ of P i(Y ). We have

Q(F (α), F (β)) = (−1)i(i−1)/2q−n∫f∗(α ∧ β ∧ ωn−i) = Q(α, β)

Moreover, since f∗ is a morphism of Hodge structures, it preserves the Weil operatorC. Therefore F is unitary with respect to Q, so its eigenvalues have norm 1. Thisgives the desired estimate on absolute values of the eigenvalues of f∗.

Since f∗ can be represented by an integer matrix, the set of it eigenvalues isa Galois invariant set of algebraic integers. So we get a stronger conclusion thatall Galois conjugates have absolute value qi/2. This would imply, that when q = 1(e.g. if f is an automorphism) then these roots of unity.

Exercises

1. Verify the above theorem for Y a complex torus, and f : Y → Y multipli-cation by a nonzero integer n.

2. Show, by example, that it no f∗ω is a multiple of ω, then the eigenvaluesof f∗ on Hi(Y ) can have different absolute values.

18.4. Conjectures of Grothendieck and Hodge

Grothendieck suggested that one should be able to carry out a proof similarto Serre’s for the final Weil conjecture. However making this work would require asolution to his standard conjectures [GSt, Kl] which are, at present, very much openeven over C. Grothendieck had formulated his conjectures in order to construct histheory of motives which gives a deeper explanation for some of the analogies betweenthe worlds of arithmetic and complex geometry. So even though Deligne managedto prove the last of Weil’s conjecture by another method, the problem of solvingthese conjectures is fundamental.

We want to spell out some of these conjectures in the complex case, and indicateits relation to a more famous conjecture due to Hodge.Let X be an n dimensionalnonsingular complex projective variety. A codimension p algebraic cycle is finiteformal sum

∑niZi, where ni ∈ Z and Zi ⊂ X a codimension p closed subvariety.

These form an abelian group Zp(X) of infinite rank. The first task is to cut itdown to a more manageable size. Given a nonsingular ι : Z → X, we defined itsfundamental class [Z] = ι!(1) ∈ H2p(X,Z). The fundamental class can be definedeven when Z has singularities. This can be done in several ways. A quick butnonelementary method is to use Hironaka’s famous theorem [Hrn] on resolution ofsingularities. This implies that there exists a smooth projective variety Z with abirational map π : Z → Z. Let i : Z → X denote the composition of π and theinclusion. Then set [Z] = i!(1) ∈ H2p(X,Z).

Lemma 18.4.1. This class is independent of the choice of resolution of singu-larities.

Proof. Let Z ′ → Z be another resolution. Then by applying Hironaka’stheorem to the fibred product, we see that there exists a third resolution Z ′′ → Z

18.4. CONJECTURES OF GROTHENDIECK AND HODGE 231

fitting into a commutative diagram.

Z ′′ψ //

Z

π

Z ′

π′ // Z

Then i!(1) = i!(ψ!1) = (i ψ)!(1). Therefore Z and Z ′′ give the same class. Bysymmetry Z ′ and Z ′′ also give the same class.

We thus get a homomorphism [ ] : Zp(X) → H2p(X,Z) by sending∑niZi 7→∑

ni[Zi]. The space of algebraic cohomology classes

H2palg(X,Z) = im[Zp(X)→ H2p(X,Z)]

H2palg(X,Q) = im[Zp(X)⊗Q→ H2p(X,Q)]

We define the space of codimension p Hodge cycles on X to be

H2phodge(X) = H2p(X,Q) ∩Hpp(X)

and let H2phodge(X,Z) denote the preimage of Hpp(X) in H2p(X,Z).

Lemma 18.4.2. H2palg(X,Z) ⊆ H2p

hodge(X,Z)

Proof. It is enough to prove that [Z] is a Hodge class. Let Z → Z be aresolution of singularities, and let i : Z → X to be the natural map. By corollary11.2.10, the map

i! : H0(Z) = Z→ H2p(X,Z)(−p)is a morphism of Hodge structures. Therefore it takes 1 takes to a Hodge class.

In more down to earth terms, this amounts to the fact that for any form α oftype (r, 2n− 2p− r) on X ∫

Z

i∗α = 0

unless r = n− p.

The Hodge conjecture asserts the converse.

Conjecture 18.4.3 (Hodge). H2palg(X,Q) and Hp

hodge(X,Q) coincide.

Note that in the original formulation Z was used in place of Q, but this isknown to be false [AH]. For p = 1, the conjecture is true – even over Z – by theLefschetz (1, 1) theorem 10.3.1. We prove it holds for p = dimX − 1.

Proposition 18.4.4. If the Hodge conjecture holds for X in degree 2p (i.e. ifH2palg(X,Q) = H2p

hodge(X,Q)) with p < n = dimX, then it holds in degree 2n− 2p.

Proof. Let L be the Lefschetz operator corresponding to a projective em-bedding X ⊂ PN . Then for any subvariety Y L[Y ] = [Y ∩ H] where H is ahyperplane section chosen in general position. It follows that Ln−2p takes H2p

alg(X)to H2n−2p

alg (X). Moreover, the map is injective by hard Lefschetz. Thus

dimH2phodge(X) = dimH2p

alg(X) ≤ dimH2n−2palg (X) ≤ dimH2n−2p

hodge (X)


On the other hand Ln−2p induces an isomorphism of Hodge structuresH2p(X,Q)(p−n) ∼= H2n−2p(X,Q), and therefore an isomorphism H2p

hodge(X) ∼= H2n−2phodge (X). This

forces equality of the above dimensions.

Corollary 18.4.5. The Hodge conjecture holds in degree 2n−2. In particularit holds for three dimensional varieties.

Given a cycle Y ∈ Zn−p(X), define the intersection number

Z · Y =∫X

[Z] ∪ [Y ] ∈ Z

This can be defined purely algebro-geometric methods over any field [F1].

Definition 18.4.6. Z ∈ Zp(X) to be homologically equivalent to 0 if [Z] = 0.It is numerically equivalent to 0 if for any Y ∈ Zn−p(X) we have Z · Y = 0. Twocycles are homologically (respectively numerically) equivalent if their difference ishomologically (respectively numerically) equivalent to 0.

Numerical equivalence is a purely algebro-geometric notion, independant of anycohomology theory. (This is clearly an issue in positive characteristic where one hasseveral equally good cohomology theories, such as the various `-adic theories.) Onthe other hand, it is usually easier to prove things about homological equivalence.For example, H2p

alg(X) which is Zp(X) modulo homological equivalence is finitelygenerated since it is sits in the finitely generate group H2p(X,Z). Therefore Zp(X)modulo numerical equivalence is also finitely generated, because clearly homologicalequivalence implies numerical equivalence. The converse is one of Grothendieck’sstandard conjectures.

Conjecture 18.4.7 (Conjecture D). Numerical equivalence coincides with ho-mological equivalence.

In order to explain the relationship to Hodge, we state another of Grothendieck’sconjectures.

Conjecture 18.4.8 (Conjecture A). The Lefschetz operator induces an iso-morphism on the spaces of algebraic cycles

Li : Hn−ialg (X,Q)→ Hn+i

alg (X,Q)

There are several other conjectures which we will not state. One of them, whichis a version of the Hodge index theorem, is true over C. The remaining conjecturesare known to follow if conjecture A is true for all X [GSt, Kl]. These conjecturesare weaker than Hodge’s and are known in many more cases. For example they areknown for all Abelian varieties, while Hodge is still open for this class.

Proposition 18.4.9. If the Hodge conjecture holds for X, then conjecture Awould also hold for it. If conjecture A holds for X then conjecture D holds for it.

Proof. By the hard Lefschetz theorem

Li : Hn−i(X,Q)→ Hn+i(X,Q)

is isomorphism of vector spaces. It follows that Li gives an isomorphism Hn−i(X) ∼=Hn+i(X)(i) of Hodge structures. Therefore it induces an isomorphism on the spacesof Hodge cycles

Hn−ihodge(X,Q)→ Hn+i

hodge(X,Q)

18.5. PROBLEM OF COMPUTABILITY 233

By the Hodge conjecture this coincides with spaces of algebraic cycles.By the Hodge index theorem 14.1.4, we get a positive bilinear form Q on Hi(X)

given by

Q(α, β) =∫X

α ∪ β′

whereβ′ =

∑±Ln−i+jβj

with β =∑Ljβj the decomposition into primitive parts in theorem 14.1.1. Suppose

that conjecture A holds forX. Then the decomposition into primitive parts restrictsto the space of algebraic cycles

H2palg(X) =

⊕LjP 2p−2j(X) ∩H2p−2j

alg (X)

Therefore β′ ∈ H2n−2palg (X) whenever β ∈ H2p

alg(X). Suppose that Z ∈ Zp(X) isnumerically equivalent to 0. Then [Z] = 0, since otherwise we get a contradictionZ · [Z]′ = Q([Z], [Z]) > 0.

Further information about the Hodge conjecture can be found in [Lw]. See[An] for an introduction to motives which has been lurking behind the scenes.

Exercises

1. Prove any of these conjectures.2. Or find counterexamples.

18.5. Problem of Computability

As we saw in this book it is relatively straightforward to compute Hodge num-bers. For things like hypersurfaces, there are formulas. More generally given explicitequations for a subvariety X ⊂ Pn, one has the following strategy for computingHodge numbers.

• View the sheaves ΩpX as coherent sheaves on Pn. Specifically:

ΩpX = ΩpPn/(IΩpPn + dI ∧ Ωp−1Pn ),

where I is the ideal sheaf of X. These sheaves can be given an explicitpresentation by combining this formula with the presentation

OPn(−p− 2)(n+1p+2) → OPn(−p− 1)(

n+1p+1) → ΩpPn

coming from corollary 16.1.3.• Resolve these as in theorem 15.3.8.• Calculate cohomology using the resolution.

This can be turned into an algorithm using standard Grobner basis techniques.(We are using the term “algorithm” somewhat loosely, but to really get one weshould assume that the coefficients are given to us in some computable subfield ofC such as Q or Q(

√2, π). )

If one wanted to verify (or disprove) the Hodge conjecture by examples, onewould run up against the following problem, which by contrast to the case of Hodgenumbers, seems extremely difficult:

Problem 18.5.1. Find an algorthim to compute the dimension of H∗alg(X,Q)and H∗hodge(X,Q) given the equations (or some other explicit representation).


In fact, the special case would already be interesting and probably very hard.

Problem 18.5.2. Find an algorithm for computing the Picard number of asurface in P3 or of a product of two curves.

We end with a few comments. Given a variety X, the Hodge structure onHk(X) is determined by the period matrices

P p =(∫

γi

ωj

)where γi is basis of Hk(X,Z) and ωj a basis of Hk−p(X,ΩpX). When X is definedover Q, we should in principle be able to compute the entries of these matrices to anydesired degree of accuracy, by combing symbolic methods with numerical ones. Butthis does not (appear to) help. However, Kontsevich and Zagier [KZ] propose thatthere may be an algorithm to determine whether or not such a number (that theycall a period) is rational. More generally, we can ask for an algorithm for decidingwhether any finite set of periods is linearly dependent over Q. Such an algorithmwould be instrumental in finding an algorithm for computing dimH∗hodge(X,Q).

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Index

1-form, 31

acyclic resolution, 69

acyclic sheaf, 69

adjunction, 160

affine space, 22

affine variety, 23

algebraic variety, 24

Betti number, 67

canonical divisor, 93, 160

Cech cohomology, 107

Chern class, 111, 185

Chow’s theorem, 203

classical topology, 22

coherent sheaf, 193

complete variety, 34

complex manifold, 20

cotangent space, 28

de Rham cohomology, 71

differential of a map, 28

Dolbeault’s theorem, 129

elliptic curve, 7, 22, 85

epimorphism, 40

equivalence of categories, 94

Euler characteristic, 67, 161

flasque sheaf, 57

flat morphism, 216

free action, 86

GAGA, 201

Grassmanian, 22

hard Lefschetz theorem, 175

harmonic form, 119

Hilbert polynomial, 200

Hodge conjecture, 231

Hodge decomposition, 136

Hodge number, 136

Hodge structure, 139

Hodge theorem, 119

Hodge-Riemann bilinear relations, 176

hypercohomology, 146

ideal sheaf, 49

Kahler identities, 131

Kahler manifold, 133

Kahler metric, 133

Lefschetz (1, 1) theorem, 138

Lefschetz pencil, 180

Lefschetz trace formula, 81

Leray cover, 109

line bundle, 31

local ring, 26

locally free, 51

locally ringed space, 27

manifold, 20

monomorphism, 40

nonsingular point, 30

p-form, 54

Picard group, 108

Picard number, 158

Poincare duality, 74

Poincare’s lemma, 72

presheaf, 18

primitive cohomology, 175

projective space, 21

projective variety, 25

properly discontinuous action, 86

quasi-isomorphism, 146

regular function, 23

Riemann surface, 20

Riemann-Roch, 94, 161

Riemannian manifold, 118

Riemannian metric, 118

ringed space, 19, 43

scheme, 45

sheaf, 18

sheaf cohomology, 59

sheaf of modules, 50

239

240 INDEX

simplicial (co)homology, 102

simplicial complex, 102

singular cohomology, 103singular point, 30

smooth morphism, 216

smooth point, 30soft sheaf, 62

stalk, 26

support, 39

tangent space, 28

tautological line bundle, 32toric variety, 48

variety, 24vector bundle, 31

vector field, 31

Zariski topology, 22

Book algebra

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