On the Homological Mirror Symmetry Conjecturefor Pairs of Pants

arXiv:1012.3238

Nick Sheridan

January 25, 2011

1

1 Overview

We consider the symplectic geometry of the smooth complex affine algebraic variety

Pn :=

n+2∑j=1

zj = 0

⊂ CPn+1 \⋃j

{zj = 0}.

This is called the n-dimensional pair of pants. It is isomorphic to CPn with n+2 generic

hyperplanes removed.

Note that P1 = CP1 \ {3 points}, i.e., the standard pair of pants.

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The pair of pants is of interest to us for three reasons:

• It is conjectured to be mirror to the Landau-Ginzburg model (Cn+2,W ), where

W = z1 . . . zn+2

(proven by Abouzaid-Auroux-Efimov-Katzarkov-Orlov for n = 1);

• Other interesting hypersurfaces are finite covers of Pn, for example the affine Fermat

hypersurface

Xn :=

n+2∑j=1

zn+2j = 0

⊂ CPn+1 \⋃j

{zj = 0}

is a cover of Pn via the map

[z1 : . . . : zn+2] 7→ [zn+21 : . . . : zn+2

n+2];

• Mikhalkin has shown that any algebraic hypersurface in (C∗)n+1 can be decomposed

into pairs of pants, analogously to the pair of pants decomposition for Riemann surfaces.

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We consider the A-model (Fukaya category) on Pn. We construct an immersed Lagrangian

sphere Ln : Sn → Pn with transverse self-intersections. Ln can be regarded as an object

of the Fukaya category.

We compute its Floer cohomology algebra:

HF ∗(Ln, Ln) ∼= Λ∗Cn+2.

4

xx

x

x

xx

1

1

2

2

3

3u

L1 : S1 → P1 is an immersed circle with three self-intersections. One can compute

HF ∗(L1, L1) ∼= C〈x1, x2, x3〉,

where x1, x2, x3 are odd super-commuting variables, so this is an exterior algebra.

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The correspondence goes like this:

generator of H0(S1) 7→ 1,

generator of H1(S1) 7→ x1 ∧ x2 ∧ x3,

xi 7→ xi,

x̄3 7→ x1 ∧ x2 etc.

The front and back triangles lead to the products µ2(x1, x2) = x̄3.

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We compute a higher product on the A∞ algebra CF ∗(Ln, Ln), which looks like this:

µn+2(e1, . . . , en+2) = ±1.

We prove that this (together with certain additional properties of the A∞ algebra coming

from the geometry) is sufficient information to determine the A∞ structure up to quasi-

isomorphism.

This allows us to prove two theorems that give some evidence for Homological Mirror

Symmetry.

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Theorem 1. The subcategory of Fuk(Pn) generated by Ln is A∞ quasi-isomorphic

to the subcategory of DbSing(W−1(0)) generated by the structure sheaf of the origin.

Recall

Xn :=

n+2∑j=1

zn+2j = 0

⊂ CPn+1 \⋃j

{zj = 0}.

Let Y n = {W = 0} ⊂ CPn+1, and Gn be the group of automorphisms of Y n given by

multiplying coordinates by (n + 2)th roots of unity.

Theorem 2. There is an A∞ embedding

PerfGn(Y n) ↪→ DπFuk(Xn)

of the category of Gn-equivariant perfect complexes of coherent sheaves on Y n into the

derived Fukaya category of Xn.

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2 Coamoebae

We need to introduce some notation:

[n + 2] := finite set {1, 2, . . . , n + 2}M̃ := rank-(n + 2) lattice Z〈e1, . . . , en+2〉eK :=

∑j∈K

ej ∈ M̃ , for K ⊂ [n + 2]

M := rank-(n + 1) lattice M̃/〈e[n+2]〉MS := M ⊗Z S for some Z-module S (usually one of R, S1,C∗).

We can identify

M̃C∗ = Cn+2 \⋃j

{zj = 0}

and the quotient

MC∗ = CPn+1 \⋃j

{zj = 0}.

We define the argument map

Arg : MC∗ → MS1.

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The closure of the image of a hypersurface in MC∗ under the Arg map is called the

coamoeba of the hypersurface.

We have

Pn =

n+2∑j=1

zj = 0

⊂MC∗.

Definition 1. Let Zn be the zonotope generated by the vectors ej in MR, i.e.,

Zn =

n+2∑j=1

θjej : θj ∈ [0, 1]

⊂MR.

Proposition 3. The coamoeba of Pn is the complement, in MS1, of the image of the

interior of Zn.

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e

e

e

1

2

3

(a) The coamoeba of P1.

e1

e2

e3

e4

(b) The coamoeba of P2.

The coamoebae of P1 and P2. The picture on the right lives in a 3-torus, drawn as a cube

with opposite faces identified, and we are removing the zonotope illustrated, which looks

somewhat like a crystal.

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Proposition 4. The map Arg is a homotopy equivalence from Pn to its coamoeba.

In particular, Pn has the homotopy type of an (n + 1)-torus with a point removed.

Corollary 5. There is a natural isomorphism

H1(Pn) ∼= M.

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3 Construction of the Lagrangian Immersion Ln : Sn → Pn

We observe that the Lagrangian L1 : S1 → P1 can be seen rather simply in the coamoeba.

It corresponds to traversing the hexagon which forms the boundary of the coamoeba.

e

e

e

x

xx

x

x x

111

2

23

3

2

3

u

Figure 1: The projection of L1 to the coamoeba.

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We will construct Ln : Sn → Pn so that Arg ◦Ln ≈ ∂Zn. To see how, we re-draw L1 in a

suggestive way:

We start with the double cover S1 → RP1 ↪→ CP1, and push it off itself with a Morse

function, so that it avoids the three divisors {zj = 0}.

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Take a Weinstein neighbourhood D∗ηRPn of RPn ⊂ CPn (for some small η > 0), and

consider its double cover D∗ηSn. We construct Ln as the graph of an exact 1-form εdf in

D∗ηSn, where f : Sn → R is a Morse function, and map it into CPn.

Think of CPn as ∑j

zj = 0

⊂ CPn+1

and Sn as ∑j

x2j = 1

⋂∑

j

xj = 0

⊂ Rn+2.

To ensure that the image of Γ(εdf ) avoids the divisors {zj = 0} in CPn, we must arrange

that ∇f is transverse to the hypersurfaces {xj = 0} in Sn.

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We define

f (x1, . . . , xn+2) :=

n+2∑j=1

g(xj),

where g : R→ R looks like this:

x

g

Figure 2: The function g.

The gradient ∇f crosses each hypersurface {xj = 0} positively. Thus the image of Γ(εdf )

avoids the divisors {zj = 0}, and defines an immersed Lagrangian sphere Ln : Sn → Pn.

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Because f (−x) = −f (x), the only points where Ln has a self-intersection are at critical

points of f .

The hypersurfaces {xj = 0} divide Sn into 2n − 2 regions, indexed by the set K of

coordinates which are negative in that region. Each region contains a unique critical point

pK of f .

pK is characterised by the property that all its negative coordinates are equal, and all its

positive coordinates are equal.

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123

12

1

13

3

23

2

24

234

34

134

14

124

4

44

1DR

2DR

3DR

4DR

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Note that Arg(pK) = eK is a vertex of Zn.

Because ∇f crosses the hypersurface {xj = 0} positively, the argument

arg(zj) ≈ arg

(xj + iε

∂f

∂xj

)∈ [0, π].

So the image of the flowline from pK to pK\{j} under the argument map is approximately

given by decreasing θj from π to 0 and keeping all other coordinates fixed.

One can show in this way that Arg ◦ Ln ≈ ∂Zn.

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4 Properties of the A∞ algebra A := CF ∗(Ln, Ln)

The generators of the A∞ algebra A = CF ∗(Ln, Ln) are

• generators of H∗(Sn), which we label pφ and p[n+2], and

• ordered pairs (p, q) of distinct points in Sn such that Ln(p) = Ln(q).

We have shown that the self-intersections of Ln are exactly the pairs (pK, pK̄) forK ⊂ [n+2]

proper and non-empty.

Thus,

A =⊕

K⊂[n+2]

C〈pK〉.

Note that

A ∼= Λ∗M̃C

as a vector space, via the isomorphism

pK 7→∧j∈K

ej.

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Homogeneity property: If the coefficient of pK0 in the A∞ product

µk(pK1, . . . , pKk)

is non-zero, then

eK0 =

k∑j=1

eKjin M

(this is because the boundary of the holomorphic disk giving this product must lift to a

closed curve in the universal cover).

Grading: One can compute that the generator pK of A has (fractional) grading given by

n|K|/(n + 2), so we also have

n

n + 2|K0| = 2− k +

n

n + 2

k∑j=1

|Kj|

.

Combining these, one can show that the only non-zero A∞ products µk are those for which

k = 2 + nq for some q ∈ Z.

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5 Flipping pearly trees

For an embedded Lagrangian L, we define CF ∗(L,L) to be generated by the critical

points of a Morse function h : L→ R. We define A∞ products by counting pearly trees,

which consist of holomorphic disks with boundary on L, connected by Morse flowlines of

h:p

1p 2

p 3

p 4

p 5

p 6 p

7

p 8

p 0

Figure 3: A pearly tree contributing to the coefficient of p0 in µ8(p1, . . . , p8).

We can think of this as pushing L off itself with the Morse function εh, then the Morse

flowlines correspond to thin strip regions in the limit ε→ 0.

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We can do the same thing for our immersed Lagrangian Ln: we have defined it by pushing

the double cover Sn → RPn ↪→ CPn off itself with the Morse function εf .

There are now two possibilities for our strip regions in the limit ε→ 0: they have boundary

on opposite sheets of the double cover (flipping strips), or they have boundary on the same

sheet (non-flipping strips).

We define a flipping pearly tree to consist of holomorphic disks with boundary on

RPn, connected by edges which can be one of two types: flipping edges (flowlines of ∇f )

or non-flipping edges (flowlines of ∇h).

A flipping pearly tree also comes with a lift of the boundary to Sn.

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It is possible to define an intersection number of a flipping pearly tree with each divisor

{zj = 0}, by reconstructing the topological class of a corresponding disk for some small

ε > 0, and taking the topological intersection number.

A flipping pearly tree is admissible if all of these intersection numbers are 0.

We define an A∞ structure by counting admissible flipping pearly trees instead of disks,

and show that this is quasi-isomorphic to A.

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We compute µ2 explicitly. If the coefficient of pK0 in µ2(pK1, pK2) is non-zero, then

eK0 = eK1 + eK2 (homogeneity), and

|K0| = |K1| + |K2| (grading),

so K0 = K1 tK2.

The coefficient is given by a count of flipping pearly trees like this:

p K 1

p K 2

p K 0

When n = 2, the flipping pearly tree giving µ2(p{2}, p{3}) = ±p{2,3} looks like this:

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Figure 4: Red = boundary of holomorphic disk, Blue = Morse flowlines of f .

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After a harrowing sign computation, it follows that

A ∼= Λ∗M̃C

as a C-algebra.

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Now we compute some higher products in A. Recall that the products µk vanish unless

k = 2 + nq, so the next product to consider is µn+2.

We compute

µn+2(p{1}, . . . , p{n+2}) = ±pφ,and the product is zero if we permute the inputs in any other way.

The corresponding flipping pearly tree is (half of) the unique degree-n curve through the

points p1, . . . , pn+2, pφ (which exists by Veronese’s theorem).

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We can parametrize our curve as u : D → CPn,

u(z) :=

−(n + 1) 1 . . . 1

1 −(n + 1) . . . 1... ... . . . ...

1 1 . . . −(n + 1)

(z − ν1)−1

(z − ν2)−1

...

(z − νn+2)−1

=

− n + 1

z − ν1+∑j 6=1

1

z − νj: − n + 1

z − ν2+∑j 6=2

1

z − νj: . . . : − n + 1

z − νn+2+∑j 6=n+2

1

z − νj

.We have

u(νj) = [1 : 1 : . . . : −(n + 1) : . . . : 1] = p{j},

and we choose νj so that u(0) = pφ.

As part of the definition of a flipping pearly tree, there must exist a lift of the boundary

to Sn which changes sheets at the points p{j} but not at pφ. We get this automatically,

because the dominant term (z − νj)−1 changes sign at νj.

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6 Versality of A

We consider the set A of A∞ algebras B with the properties

• µ1 = 0;

• (B, µ2) ∼= Λ∗M̃C as a C-algebra;

• B has the same homogeneity and grading properties as A,

modulo quasi-isomorphism.

A Hochschild cohomology computation shows that the deformation space of A is 1-dimensional,

and our computation of µn+2 shows exactly that the deformation class of A is non-trivial

(A is versal). Any two versal elements are quasi-isomorphic.

Thus, we have determined A up to quasi-isomorphism.

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By a theorem of Orlov, DbSing(W−1(0)) is quasi-equivalent to MF (R,W ), the category

of matrix factorisations of W over the ring R := C[z1, . . . , zn+2].

Dyckerhoff has shown how to compute a minimal A∞ model for the endomorphism algebra

of the stabilised residue field, using the Homological Perturbation Lemma.

One can show that this A∞ algebra satisfies the required properties to lie in A. Dyckerhoff

also computed its deformation class, which is non-trivial. Hence it is quasi-isomorphic to

A.

This completes the proof of Theorem 1.

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7 Covers of Pn

Let X → Pn be a finite covering with covering group Γ. Let L̃n denote the direct sum of

lifts of Ln to X . Then

CF ∗(L̃n, L̃n) ∼= CF ∗(Ln, Ln) o Γ∗,

where Γ∗ is the character group of Γ.

One very interesting case is the affine Fermat hypersurface:

Xn := {zn+21 + . . . + zn+2

n+2 = 0} ⊂ CPn+1 \⋃j

{zj = 0},

with the covering Xn → Pn given by

[z1 : . . . : zn+2] 7→ [zn+21 : . . . : zn+2

n+2]

with covering group Γn = M ⊗ Zn+2.

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The lifts of Ln form a collection of (n+ 2)n+1 embedded Lagrangian spheres in Xn, whose

endomorphism algebra is

Ao Γ∗n.

This allows us to prove Theorem 2.

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