UNIFORMIZATION BY LAURICELLA FUNCTIONS—AN OVERVIEW … · 2. Orbifolds and discrete monodromy...

arX

iv:m

ath/

0507

534v

1 [m

ath.

CV

] 26

Jul

200

5 UNIFORMIZATION BY LAURICELLA FUNCTIONS—AN OVERVIEW OFTHE THEORY OF DELIGNE-MOSTOW

EDUARD LOOIJENGA

ABSTRACT. This is a survey of the Deligne-Mostow theory of Lauricellafunctions, orwhat almost amounts to the same thing, of the period map for cyclic coverings of theRiemann sphere.

INTRODUCTION

These notes are about the theory of hypergeometric functions in several variables. Thefunctions in question generalize the Gauß hypergeometric function and are obtained asintegrals of a multivalued differential of the form

ηz := (z0 − ζ)−µ0 · · · (zn − ζ)−µndζ,

wherez0, . . . , zn are pairwise distinct complex numbers (and are allowed to vary) andthe exponentsµk are taken in the open unit interval(0, 1) (and are always kept fixed). Ifγ is a path connecting somezk with somezl whose relative interior avoids thezk’s andif a determination of the differential along that path is chosen, thenηz can be integratedalongγ (the integral will indeed converge). That integral will depend holomorphically onz = (z0, . . . , zn), for if we vary z a little, then we can letγ and the determination ofηzfollow this variation in a continuous manner. The (multivalued) function ofz thus obtainedis the type of hypergeometric function that takes the stage here. We now briefly explainwhich are the aspects of particular interest that will make an appearance in this piece.

One readily finds that it is better not to focus on one such integral, but to consider allof them simultaneously, or rather, to consider for everyz as above (and fixed exponents),the spaceLz of power series expansions inn + 1 complex variables atz that are linearcombinations of such integrals. It turns out that this vector spaceLz has dimensionnand that the ‘tautological’ map-germ(Cn+1, z) → L∗

z sendsz to an element6= 0 andhas the following regularity property: ifM0,n+2 stands for the configuration space of(n+1)-tuples inC modulo affine-linear equivalence (which is also the configuration spaceof (n + 2)-tuples on the Riemann sphere modulo projective-linear equivalence), then thismap-germ drops to a local isomorphism(M0,n+2, [z]) → P(L∗

z). By analytic continuationwe have an identification ofLz with Lz′ for nearbyz′ and the multivalued nature of thehypergeometric functions is reflected by the fact that if we letz traverse a loop in the spaceof pairwise distinct(n+1)-tuples and if let the elements ofLz follow that loop by analyticcontinuation, then there results a linear (monodromy) transformation ofLz which need notbe the identity. The transformations ofL∗

z thus obtained form a subgroupΓ of GL(L∗z),

called the monodromy group of the system. The main questionsaddressed here are:1. When doesΓ leave invariant a Hermitian form which is a positive definite, semidefi-

nite or of hyperbolic signature?

2000Mathematics Subject Classification.Primary: 33C65, 22E40; Secondary: 32G20.Key words and phrases.Lauricella function, ball quotient.

1

http://arxiv.org/abs/math/0507534v1

2 EDUARD LOOIJENGA

2. In the situation of Question 1, when isΓ discrete as a subgroup ofGL(L∗z)? (This

is essentially equivalent to: when actsΓ properly onD?) And when isΓ arithmetic (in anaturally definedQ-algebraic group that containsΓ)?

The answer to the first question is short enough to give here: whenµ0+ · · ·+µn is< 1,= 1 or in the interval(1, 2) respectively (we are not claiming the converse). In that caseM0,n+2 acquires a metric of constant holomorphic curvature as follows. First, we observethatP(L∗

z) contains a complex symmetric manifold of constant holomorphic curvatureDas an open subset on whichΓ acts by isometries: we get respectively all ofP(L∗

z) withits Fubini-Study metric, an affine space inP(L∗

z) with a translation invariant metric or anopen ball with its complex hyperbolic metric. But we also findthat the local isomorphism(M0,n+2, [z]) → P(L∗

z) lands inD, so thatM0,n+2 inherits a metric fromD.Question 2 is harder. IfΓ is discrete as well, then the exponentsµk must be rational

numbers. One of the main results states thatM0,n+2 has then finite invariant volume andthat its natural metric completion is an algebraic variety (we get a projective space in theelliptic and parabolic cases and in the hyperbolic case it isobtained by adding the stableorbits in a setting of geometric invariant theory). Deligneand Mostow gave sufficientconditions for discreteness, which were later weakened by Mostow and Sauter to makethem sufficient as well.

If the µk ’s are rational, then there is the connection with the theoryof period maps(regardless whetherΓ is discrete): ifm is their smallest common denominator and if wewrite µk = dk/m, then the hypergeometric functions become periods of the cyclic coverof C defined bywm = (z0 − ζ)d0 · · · (zn − ζ)dn . Forηz then lifts to a regular univalueddifferential on this affine curve (regular resp. with simplepoles at infinity when

∑k µk

is greater than resp. equal to1) andγ is covered by a cycle such that the hypergeometricintegral is the period of the lift over this cycle.

As the reader will have gathered, this is mostly an account ofwork of Mostow (andhis student Sauter) and of Deligne-Mostow. It is self-contained in that the sense that wehave included proofs (except for a technical lemma needed for an arithmeticity criterion).Occasionally our treatment somewhat differs from theirs. For instance, our discussion ofinvariant Hermitian forms does not use the approach in [5] inspired by Hodge theory, butrather follows the more pedestrian path in [3]. We also foundit natural to use the languageof orbifolds throughout. For some of the history of the material expounded here, we referto the first and the last section of [5] as well to the review [2]. In Section 5 we—verysketchily—mention some recent developments.

This paper is based on a series of talks I gave at the CIMPA summer school (2005)in Istanbul. I thank my hosts, in particular Professor Uludag, for their hospitality and formaking this summer school such a pleasant and fruitful experience.

CONTENTS

Introduction 11. The Lauricella differential 31.1. Definition and first properties 31.2. Lauricella arc systems 41.3. The rank of the Schwarz map 61.4. When points coalesce 71.5. Monodromy group and monodromy cover 81.6. Invariant Hermitian forms 91.7. Cohomological interpretation via local systems of rank one 12

UNIFORMIZATION BY LAURICELLA FUNCTIONS 3

2. Orbifolds and discrete monodromy groups 132.1. Monodromy defined by a simple Dehn twist 132.2. Extension of the evaluation map 132.3. The elliptic and parabolic cases 143. The hyperbolic case 173.1. A projective set-up 173.2. Extending the range of applicability 194. Modular interpretation 204.1. Cyclic covers ofP1 204.2. Arithmeticity 224.3. Working over a ring of cyclotomic integers 235. Generalizations and other view points 255.1. Higher dimensional integrals 255.2. Geometric structures on arrangement complements 25References 27

1. THE LAURICELLA DIFFERENTIAL

1.1. Definition and first properties. Assume given real numbersµ0, . . . , µn in the in-terval (0, 1), wheren > 0. We shall refer to the(n + 1)-tupleµ = (µ0, . . . , µn) as aweight systemand we call its sum|µ| := ∑n

i=0 µi the total weightof µ. TheLauricelladifferentialof weightµ is

ηz := (z0 − ζ)−µ0 · · · (zn − ζ)−µndζ, with z = (z0, . . . , zn) ∈ (Cn+1).

(Here(Cn+1) stands for the set of(z0, . . . , zn) ∈ Cn+1 whose components are pairwisedistinct.) Athough this differential is multivalued, it has a natural determination on a lefthalf space by taking there the value of the integrand whose argument is< π/|µ| in absolutevalue. We further note thatηz is locally integrable as a multivalued function: nearzk, ηzis of the form(ζ − zk)

−µk exp(holom)dζ; this is the differential of a function of the formconst+ (ζ − zk)

1−µk exp(holom) and since1 − µk > 0, that function takes a well-defined value inzk. This implies thatηz can be integrated along everyrelative arcof(C, z0, . . . , zn); by the latter we mean an oriented piecewise differentiablearc in Cwhose end points lie inz0, . . . , zn, but which does not meet this set elsewhere.

The behavior of the differential at infinity is studied by means of the substitutionζ =ω−1; this gives

ηz = −(ωz0 − 1)−µ0 · · · (ωzn − 1)−µnω|µ|−2dω,

which suggests to putzn+1 := ∞ andµn+1 := 2 − |µ|. In caseµn+1 < 1 (equivalently,|µ| > 1), ηz is also (multivalued) integrable atzn+1.

Remark1.1. Following Thurston [14], we may think ofηz as a way of putting a flat Eu-clidean structure onP1 with singularities atz0, . . . , zn+1: a local integral ofηz defines ametric chart with values inC, but now regarded as the Euclidean plane (so the associatedmetric is simply|ηz |2). At zk, k ≤ n, the metric space is isometric to a Euclidean conewith total angle2π(1 − µk); this is also true fork = n+ 1 in caseµn+1 < 1, or equiva-lently, |µ| > 1; if |µ| = 1 resp.|µ| < 1, then a punctured neighborhood of∞ is isometricto a flat cylinder resp. the complement of a compact subset of aEuclidean cone with totalangle1− |µ|.

4 EDUARD LOOIJENGA

Let be given relative arcγz of (C, z0, . . . , zn) and let also be given a determinationof ηz on γz so that

∫γz

η is defined. Choose an open disksDk aboutzk in C such thattheD0, . . . , Dn are pairwise disjoint. Then we can find for everyz′ ∈ D0 × · · · × Dn,a relative arcγz′ of (C, z′0, . . . , z′n) and a determination ofηz′ on supp(γz′) such thatboth depend continuously onz′ and yield the prescribed value forz = z′. Any primitiveof η near(z, zk) with respect to its second variable is (as a function of(z′, ζ)) of the formg(z′) + (ζ − z′k)

1−µkh(ζ, z′), with g andh holomorphic and so the function

z′ ∈ D0 × · · · ×Dn 7→∫

γz′

ηz′ ∈ C

is holomorphic. We call such a function (or some analytic extension of it) aLauricellafunction. The Lauricella functions (with given weight systemµ) span a complex vectorspace. We denote the space of germs of holomorphic functionsat z ∈ (Cn+1) thatare germs of Lauricella functions byLz. It is clear that forz′ ∈ D0 × · · · × Dn, wecan naturally identifyLz′ with Lz. Here are some elementary properties of Lauricellafunctions (the proofs are left to the reader, who should be duely careful with exchangingdifferention and integration in the proof of (c) ).

Proposition 1.2. Anyf ∈ Lz

(a) is translation invariant:f(z0 + a, . . . , zn + a) = f(z0, . . . , zn) for smalla ∈ C,(b) is homogeneous of degree1−|µ|: f(etz0, . . . , etzn) = e(1−|µ|)tf(z0, . . . , zn) for

smallt ∈ C and(c) obeys the system of differential equations

∂2f

∂zk∂zl=

1

zk − zl

(µl

∂

∂zk− µk

∂f

∂zl

), 0 ≤ k < l ≤ n.

The translation invariance of the Lauricella functions suggests to introduce

Vn := Cn+1/main diagonal and V n := (Cn+1)/main diagonal,

as they are in fact defined onV n . The homogeneity implies that when|µ| = 1, these

functions are also constant on theC×-orbits and hence factor throughP(V n ); for reasons

which will become clear later, we call this theparaboliccase.An important consequence of part (c) of the preceding proposition is

Corollary 1.3. The map which assigns tof ∈ Lz its 1-jet atz is injective.

Proof. If f ∈ Lz, then its partial derivativesfk := ∂f∂zk

satisfy the system of ordinarydifferential equations

∂fk∂zl

=1

zk − zl(µlfk − µkfl) , k 6= l.

We can complete this system as to get also such equations for∂fk∂zk

by using the fact∑k fk = 0 (which follows from the translation invariance). The elementary theory of

such systems says that there is precisely one solution for it, once the initial conditionsfk(z) are prescribed. To such a solution corresponds at most one element ofLz up to aconstant.

1.2. Lauricella arc systems.

Definition 1.4. Given(z0, . . . , zn) ∈ Cn+1, we define anL-arc systemas to be an orientedarc in the Riemann sphereP1 = C∪∞ from z0 to zn+1 = ∞ which passes successivelythroughz1, . . . zn and follows near∞ the real axis in the positive direction. Ifδ is such an


L-arc system, then we denote the piece connectingzk−1 with zk by δk and we often letδalso stand for the system of arcs(δ1, . . . , δn+1).

The complement of the support ofδ is simply connected and so we have a well-defineddetermination ofηz on this complement which extends the one we already have on a lefthalf space. We also extendηz to the support ofδ itself by insisting thatηz be continuous‘from the left’ (which makes the determination ofηz discontinuous alongδ). With theseconventions,

∫δk

ηz has fork = 1, . . . , n a well defined meaning (and also makes sensefor k = n + 1 in caseµn+1 < 1). If we let z vary in a small neighborhood, we getan element ofLz that we simply denote by

∫δk

η. We denote byδ−k the arc connectingzk−1 with zk that is ‘infinitesimally’ to the right ofδk. By this we really mean thatηzis given onδ−k the determination it gets as a limit from the right. Notice that ηz |δ−k =

exp(−2π√−1(µ0 + · · ·+ µk−1))ηz |δk.

Theorem 1.5. The functions∫δ1η, . . . ,

∫δn

η define a basis forLz. Moreover,Lz containsthe constant functions if and only if we are in the parabolic case:|µ| = 1.

Proof. Any relative arc of(C, z0, . . . , zn) is homotopic to a composite of the arcsδk,and their inverses (we want the homotopy be such that the determination ofη varies con-tinuously). Since any two determinations ofη differ by a constant factor, this implies thatthe functions

∫δ1η, . . . ,

∫δn

η generateLz.If |µ| = 1, thenηz is near∞ equal to−ζ−1dζ. So then for a loopγ which encircles

z0, . . . , zn in the clockwise direction, we have∫

γ

ηz =

∫

γ

−ζ−1dζ = 2π√−1,

which proves thatLz contains the constant2π√−1.

It remains to show that ifa1, . . . , ak, c ∈ C are such that∑n

k=1 ak∫δk

η = c, thenc 6= 0

implies|µ| = 1 andc = 0 implies that allai vanish as well. We prove this with inductionon n. To this end, we consider a curvez(s) in (Cn+1) of the form(z0, . . . , zn−2, 0, s),with s > 0 and anL-arc systemδ(s) for z(s) with δ1, . . . , δn−1 fixed andδn = [0, s].By analytic continuation we may assume that

∑n−1k=1 ak

∫δk

ηz(s) + an∫ s

0ηz(s) = c. We

multiply this identity withsµn and investigate what happens fors → ∞. Fork < n,

sµn

∫

δk

ηz(s) =

∫

δk

(z0 − ζ)−µ0 · · · (zn−2 − ζ)−µn−2(−ζ)−µn−1(1− s−1ζ)−µndζ,

which fors → ∞ tends to∫δk

ηz′ , wherez′ = (z0, . . . , zn−1). On the other hand,∫ s

0

(z0 − ζ)−µ0 · · · (−ζ)−µn−1(s− ζ)−µndζ

= s(−s)−|µ|∫ 1

0

(−s−1z0 + ζ)−µ0 · · · (ζ)−µn−1(−1 + ζ)−µndζ

= s(−s)−|µ| + o(|s|1−|µ|), s → ∞.

So we find that

sµn

(c+ an

((−s)1−|µ| + o(|s|1−|µ|

))=

n−1∑

k=1

ak

∫

δk

ηz′ , s → ∞.

This shows thatc 6= 0 implies |µ| = 1 (andan = (−1)−|µ|). Suppose nowc = 0. Ifµn < |µ| − 1, then the left hand side tends to zero ass → ∞ and so the right hand side

6 EDUARD LOOIJENGA

must be zero. Our induction hypothesis then implies thata1 = · · · = an−1 = 0 and fromthis we see thatan = 0, too. If µn > |µ| − 1, then we clearly must havean = 0 and theinduction hypothesis implies thata1 = · · · = an−1 = 0, also.

Remark1.6. So the space of solutions of the system of differential equations in Proposition1.2-c is in the nonparabolic case equal toLz ⊕ C, and containsLz as a hyperplane in theparabolic case.

1.3. The rank of the Schwarz map. We find it convenient to modify our basis of Lauri-cella functions by a scalar factor by putting

Fk(z, δ) :=

∫

δk

(ζ − z0)−µ0 · · · (ζ − zk−1)

−µk−1(zk − ζ)−µk · · · (zn − ζ)−µndζ

=wk

∫

δk

ηz , wherewk := e√−1π(µ0+···+µk−1).

The notation now also displays the fact the value of the integral depends on the wholeL-arc system (which was needed to makeηz univalued) and not just onδk. Notice that ifz = x is real andx0 < x1 < · · · < xn andδ consists of real intervals, then the integrandis real valued and positive and hence so isFk. Let us also observe that

∫

δk

ηz = wkFk(z, δ) and∫

δ−k

ηz = wkFk(z, δ),

where the second identity follows from the fact thatηz |δ−k = w2kηz |δk. So if we are in the

parabolic case, then the integral ofηz along a clockwise loop which enclosesz0, . . . , znyields the identity

∑nk=1(wk − wk)Fk(z, δ) = 2π

√−1, or equivalently,

(1.1)n∑

k=1

Im(wk)Fk(z, δ) = π.

In other words,F = (F1, . . . , Fn) then maps to the affine hyperplaneAn−1 in Cn definedby this equation.

Corollary 1.7. If we are not in the parabolic case, thenF = (F1, . . . , Fn), viewedas a multivalued map fromV

n to Cn, is a local isomorphism and never takes the ori-gin of Cn as value. In the parabolic case,F = (F1, . . . , Fn) factors through a local(multivalued) isomorphism fromP(V

n ) to the affine hyperplaneAn−1 in Cn defined by∑nk=1 Im(wk)Fk = π.

Proof. Given (z, δ), consider then covectorsdF1(z, δ), . . . , dFn(z, δ) in the cotangentspace ofz. According to corollary 1.3, a linear relation among them must arise from alinear relation among the function germsF1, . . . , Fn ∈ Lz and the constant function1.According to Theorem 1.5, such a relation exists if and only if |µ| = 1. The corollaryeasily follows, except perhaps the claim thatF is nowhere zero. But ifFk(z, δ) = 0 forall k, then we must have|µ| 6= 1; sinceF will be constant zero on theC×-orbit throughzthis contradicts the fact thatF is a local isomorphism.

Definition 1.8. We call the multivalued mapF from V n to Cn theLauricella mapand its

projectivizationPF fromP(V n ) to Pn−1 theSchwarz mapfor the weight systemµ.

The above corollary tells us that the Schwarz map always is a local isomorphism (whichin the parabolic case takes values in the affine openAn−1 ⊂ Pn−1).


1.4. When points coalesce.We investigate the limiting behavior ofF when some ofthe zk ’s come together. To be specific, fix0 < r < n and let for0 < ε < 1, zε =(εz0, . . . εzr, zr+1, . . . zn) and see what happens whenε → 0. We assume here thatz1, . . . , zr lie inside the unit disk, whereas the others are outside thatdisk and chooseδaccordingly:δ1, . . . , δr resp.δr+2, . . . , δn+1 lie inside resp. outside the unit disk.

Putµ′ := (µ0, . . . , µr), z′ = (z0, . . . , zr). Then

Fk(zε, δ) = wk

∫

δk

(εz0 − ζ)−µ0 · · · (εzr − ζ)−µr (zr+1 − ζ)−µr+1 · · · (zn − ζ)−µndζ

= ε1−|µ′|wk

∫

δk

(z0 − ζ)−µ0 · · · (zr − ζ)−µr (zr+1 − εζ)−µr+1 · · · (zn − εζ)−µndζ,

where in the last line (involving the passage toεζ as the new integration varable)δk mustbe suitably re-interpreted. So fork ≤ r,

(1.2) ε|µ′|−1Fk(zε, δ) = (1 +O(ε))z

−µr+1

r+1 · · · z−µn

n F ′k(z

′, δ′),

whereF ′k is a component of the Lauricella map with weight systemµ′:

F ′k(z

′, δ′) = wk

∫

δk

(z0 − ζ)−µ0 · · · (zk − ζ)−µkdζ.

If k > r and in casek = r + 1, |µ′| < 1, we find

(1.3) Fk(zε, δ) = (1 +O(ε))wk

∫

δk

(−ζ)−|µ′|(zr+1 − ζ)−µr+1 · · · (zn − ζ)−µndζ.

Assume now|µ′| < 1. Then these estimates suggest to replace inF = (F1, . . . , Fn),for k ≤ r , Fk by ε|µ

′|−1Fk(z, δ). In geometric terms, this amounts to enlarging thedomain and range ofF : now view it as a multivalued map defined an open subset of theblowupBl(z0,...,zr) Vn of the diagonal definedz0 = · · · = zr and as mapping to the blowupBl(F1,...,Fr) C

n of the subspace ofCn defined byF1 = · · · = Fr = 0. It maps the excep-tional divisor (defined byε = 0) to the exceptional divisorPr−1×Cn−r ⊂ Bl(F1,...,Fr)C

n.If we identify the exceptional divisor in the domain withP(Vr)×V1+n−r (the second com-ponent begins with the common value ofz0, . . . , zr), then we see that the first componentof this restriction is the Schwarz mapPF ′ for the weight systemµ′ and the second com-ponent iswr times the Lauricella map for the weight system(µ′, µr+1, . . . , µn).

If several such clusters are forming, then we have essentially a product situation.We shall also need to understand what happens when|µ′| = 1. Then taking the limit for

ε → 0 presents a problem forFr+1 only (the other components have well-defined limits).This is related to the fact thatηz is univalued on the unit circleS1; by the theory of residueswe then have∫

S1

ηz =

∫

S1

(z0 − ζ)−µ0 · · · (zn − ζ)−µndζ = 2π√−1z

−µr+1

r+1 · · · z−µn

n .

We therefore replaceηz by ηz := zµr+1

r+1 · · · zµn

n ηz andF by F := zµr+1

r+1 · · · zµn

n F . Thisdoes not change the Schwarz map, of course. Notice however, that now

∫S1 ηz = 2π

√−1.

Lemma 1.9. Assume thatµ′ is of parabolic type:|µ′| = 1. Define Lauricella dataµ′′ :=(µr+1, . . . , µn+1), z′′ := (z−1

r+1, . . . , z−1n , 0) and letδ′′ = (δ′′1 , . . . , δ

′′n−r) be the image of

(δr+2, . . . , δn+1) under the mapz 7→ z−1. Then we have

Fk(zε, δ) =

(1 +O(ε))F ′

k(z′, δ′) when1 ≤ k ≤ r,

(1 +O(ε))F ′′k−r−1(z

′′, δ′′) whenr + 2 ≤ k ≤ n,

8 EDUARD LOOIJENGA

whereaslimε→0 Re Fr+1(zε, δ) = +∞. Moreover,∑r

k=1 Im(wk)Fk(z, δ) = π.

Proof. The assertion fork ≤ r is immediate from our previous calculation. For1 ≤ i ≤n− r − 1 we find

Fr+1+i(zε, δ) =

= wr+1+i

∫

δr+1+i

(εz0 − ζ)−µ0 · · · (εzr − ζ)−µr (1− ζ

zr+1)−µr+1 · · · (1− ζ

zn)−µndζ

= −w′′i

∫

δ′′i

(εz0− ζ−1)−µ0 · · · (εzr − ζ−1)−µr (1− 1

ζzr+1)−µr+1 · · · (1− 1

ζzn)−µn

dζ

−ζ2

= w′′i

∫

δ′′i

(1−εz0ζ)−µ0 · · · (1−εzrζ)

−µr (1

zr+1−ζ)−µr+1 · · · ( 1

zn−ζ)−µn(−ζ)−µn+1dζ

= (1 +O(ε))F ′′i (z

′′, δ′′).

As to the limiting behavior ofFr+1, observe that

Fr+1(zε, δ) =

= −∫

δr+1

(εz0 − ζ)−µ0 · · · (εzr − ζ)−µr (1− ζ

zr+1)−µr+1 · · · (1 − ζ

zn)−µndζ

=

∫

δr+1

(ζ − εz0)−µ0 · · · (ζ − εzr)

−µr (1− ζ

zr+1)−µr+1 · · · (1− ζ

zn)−µndζ.

For ε → 0, the integrand tends toζ−1(1 − ζ/zr+1)−µr+1 · · · (1 − ζ/zn)

−µn , from whichthe asserted limiting behaviour easily follows. The last assertion follows from the fact that∫S1 ηz = 2π

√−1 (see the derivation of Equation (1.1)).

So if we regard the Schwarz map as defined on an open subset ofBl(z0,...,zk) P(Vn),then its composite with the projection ofPn → Pn−1 obtained by omittingFr is on theexceptional divisor given by[F ′

1 : · · · : F ′r : F ′′

1 : · · · : F ′′n−1−r].

1.5. Monodromy group and monodromy cover. We begin with making a few remarksabout the fundamental group of(Cn+1). We take[n] = (0, 1, 2, . . . , n) as a base pointfor (Cn+1) and use the same symbol for its image inV

n . The projection(Cn+1) → V n

induces an isomorphism on fundamental groups:π1((Cn+1), [n]) ∼= π1(Vn , [n]). This

group is known as thepure(also calledcolored) braid group withn+1 strands; we denoteit by PBrn+1. Another characterization ofPBrn+1 is that as the group of connected com-ponents of the group of diffeomorphismsC → C that are the identity outside a compactsubset ofC and fix eachzk.

If α is a path in(Cn+1) from z to z′, and if we are given anL-arc systemδ for z, thenwe can carry that system continuously along when we traverseα; we end up with anL-arcsystemδ′ for z′ and thisL-arc system will be unique up to isotopy. In this wayPBrn+1

acts on the set of isotopy classes ofL-arc systems. It is not hard to see that this actionis principal: for every ordered pair of isotopy classes ofL-arc systems, there is a uniqueelement ofPBrn+1 which carries the first one onto the second one.

The groupPBrn+1 has a set of distinguished elements, calledDehn twists, defined asfollows. The basic Dehn twist is a diffeomorphism of the annulusD1,2 ⊂ C : 1 ≤ |z| ≤ 2;it is defined byre

√−1θ 7→ re

√−1(θ+φ(r)), whereφ is a differentiable function which is

zero resp.2π on a neighborhood of1 resp.2 (all such diffeomorphisms ofD1,2 are isotopicrelative to the boundary∂D1,2). If S is an oriented surface, and we are given an orientation


preserving diffeomorphismh : D1,2 → S, then the Dehn twist on the the image and theidentity map on its complement define a diffeomorphism ofS, which is also called a Dehntwist. Its isotopy class only depends on the isotopy class ofthe image of the counterclockwise oriented unit circle (as an oriented submanifoldof S). These embedded circlesoccur here as the isotopy classes of embedded circles inC−z1, . . . , zn. A particular caseof interest is such a circle encloses precisely two points ofz1, . . . , zn, sayzk andzl. Theisotopy class of such a circle defines and is defined by the isotopy class of an unorientedpath inC− z1, . . . , zn that connectszk andzl (the boundary of a regular neighborhoodof such a path gives an embedded circle). The element of the pure braid group associatedto this is calledsimple; if we choose for every pair0 ≤ k < l ≤ n a simple element, thenthe resulting collection of simple elements is known to generatePBrn+1.

There is a standard way to obtain a covering ofV n on whichF is defined as a univalued

map. Let us recall this in the present case. First notice thatif α is a path in(Cn+1)

from z to z′, then analytic continuation along this path gives rise to anisomorphism ofvector spacesρµ(α) : Lz → Lz′ . This is compatible with composition: ifβ is a path in(Cn+1) from z′ to z′′, thenρµ(β)ρµ(α) = ρµ(βα) (we use the functorial convention forcomposition of paths:βα meansα followed byβ). A loop in (Cn+1) based at[n] definesan elementρµ(α) ∈ GL(L[n]) and we thus get a representationρµ of PBrn+1 in L[n].The image of thismonodromy representationis called the themonodromy group(of theLauricella system with weight systemµ); we shall denote that group byΓµ, or simply byΓ. The monodromy representation defines aΓ-coveringV

n of V n on which theFk ’s are

univalued. A point ofV n can be represented as a pair(z, α), whereα is a path inCn+1

from [n] to z, with the understanding that(z′, α′) represents the same point if and only ifz − z′ lies on the main diagonal (so thatLz′ = Lz) andρµ(α) = ρµ(α

′). The action ofΓon V

n is then given as follows: ifg ∈ Γ is represented by the loopαg in Cn+1 from [n],theng.[(z, α)] = [(z, αα−1

g )]. But it is often more useful to represent a point ofV n as a

pair(z, δ), whereδ is anL-arc system forz, with the understanding that(z′, δ′) representsthe same point if and only ifz− z′ lies on the main diagonal andFk(z, δ) = Fk(z

′, δ′) forall k = 1, . . . , n. For this description we see right away that the basic Lauricella functionsdefine a univalued holomorphic map

F = (F1, . . . Fn) : V n → Cn.

Since[(z, δ)] only depends on the isotopy class ofδ, the action ofΓ is also easily explicatedin terms of the last description. The germ ofF at the base point defines an isomorphismL∗[n]

∼= Cn: c = (c1, . . . , cn) ∈ Cn defines the linear form onLz which sendsFk to ck. Ifwe letΓ act onCn accordingly (i.e., as the dual ofL[n]), thenF becomesΓ-equivariant.

TheC×-action onV n given by scalar multiplication will lift not necessarily toa C×-

action onV n , but to one of a (possibly) infinite coveringC×. For this action,F is homo-

geneous of degree1− |µ|. Let us denote byP(V n ) theC×-orbit space ofV

n .

1.6. Invariant Hermitian forms. Our goal is to prove the following theorem.

Theorem 1.10.If |µ| < 1, then the monodromy groupΓ leaves invariant a positive definiteHermitian formH onCn.

If |µ| = 1 (the parabolic case), thenΓ leaves invariant a positive definite HermitianformH on the (linear) translation hyperplane of the affine hyperplaneAn−1 in Cn, definedby∑n

k=1 Im(wk)Fk = 0.

10 EDUARD LOOIJENGA

If 1 < |µ| < 2, then the monodromy groupΓ leaves invariant a hyperbolic HermitianformH onCn (so of signature(n − 1, 1)) with the property thatH(F (z, z)) < 0 for allz ∈ V

n .

Before we begin the proof, let us make the following observation. If W is a finitedimensional complex vector space, then by definition a pointu of P(W ) is given bya one-dimensional subspaceLp ⊂ W . An exercise shows that the complex tangentspaceTpP(W ) of P(W ) at p is naturally isomorphic toHom(Lp,W/Lp). If we are alsogiven a Hermitian formH on W which is nonzero onLp, then it determines a Hermit-ian form Hp on TpP(W ) ∼= Hom(Lp,W/Lp) as follows: sinceH is nonzero onLp,theH-orthogonal complementL⊥

p maps isomorphicallyW/Lp; if we choose a genera-tor u ∈ Lp and think of a tangent vector as a linear mapφ : Lp → L⊥

p , then we putHp(φ, φ

′) := |H(u, u)|−1H(φ(u), φ′(u)). This is clearly independent of the generatoru.It is also clear thatHp only depends on the conformal equivalence class ofH : it does notchange if we multiplyH by a positive scalar.

If H is positive definite, then so isHp for everyp ∈ P(W ). In this wayP(W ) acquiresa Hermitian metric, known as theFubini-Studymetric. It is in fact a Kahler manifold onwhich the unitary group of(W,H) acts transitively.

There is another case of interest, namely whenH has hyperbolic signature: if we restrictourselves to the setB(W ) of p ∈ P(W ) for whichH is negative onLp, thenHp is positivedefinite as well. This defines a metric onB(W ) which is invariant under the unitary groupof (W,H). If we choose a basis of linear formsu0, . . . , um onW such thatH takes thestandard formH(u, u) = −|u0|2 + |u1|2 + · · ·+ |um|2, then we see thatB(W ) is definedin P(W ) by the inequality|u1/u0|2 + · · · + |um/u0|2 < 1, which is simply the openunit ball in complexm-space. We callB(W ) a complex-hyperbolic spaceand the metricdefined above, thecomplex-hyperbolic metric. As in the Fubini-Study case, this metricmakesB(W ) a Kahler manifold on which the unitary group of(W,H) acts transitively.Form = 1 we recover the complex unit disk with its Poincare metric.

Returning to the situation of Theorem 1.10, we see that in allthree casesPF is a localisomorphism mapping to a homogeneous Kahler manifold: when |µ| < 1, the range is aFubini-Study spacePn−1 (this notatation is a private one: the subscript is supposedto dis-tinguish it from the metricless projective spacePn−1), for |µ| = 1 we get a complex affinespace with a translation invariant metric (indeed, denotedhere byAn−1) and when|µ| > 1we get a complex ballBn−1 with its complex-hyperbolic metric. Since these structuresareΓ-invariant, we can state this more poignantly: the weight systemµ endowsP(V

n )with a natural Kahler metric locally isometric with a Fubini-Study metric, a flat metric ora complex-hyperbolic metric. We will therefore use the corresponding terminology for thecases|µ| < 1 and1 < |µ| < 2 and call them theelliptic andhyperboliccase, respectively.

Theorem 1.10 follows from more specific result that takes a bit of preparation to for-mulate. We shall associate to the weight systemµ a Hermitian formH on Cn or on thehyperplane inCn defined by

∑nk=1 Im(wk)Fk = 0 in Cn, depending on whether|µ| is

integral. We do this somewhat indirectly. LetH be the Hermitian form onCn+1 definedby

H(F,G) =∑

1≤j<k≤n+1

Im(wjwk)FkGj .

TheH-orthogonal complement inCn+1 of the last basis vectoren+1 is the hyperplaneAdefined by

∑n+1k=1 Im(wk)Fk = 0. When|µ| 6∈ Z, the projectionA ⊂ Cn+1 → Cn (which

forgets the last coordinate) is an isomorphism (sincewn+1 = eπ√−1|µ|, Im(wn+1) 6= 0 in


that case) and thus identifiesCn with this A; we letH then be the restriction ofH to Cn.If |µ| ∈ Z, thenIm(wn+1) = 0 and hence the projectionA ⊂ Cn+1 → Cn has kernelCen+1 and image the hyperplaneA in Cn defined by

∑nk=1 Im(wk)Fk = 0. So thenH

induces a Hermitian form onA. The following proposition implies Theorem 1.10.

Proposition 1.11. The formH is Γ-invariant for all weight systemsµ. For 0 < |µ| ≤ 1,the formH is positive definite. For1 < |µ| < 2, H is of hyperbolic signature and we haveH(F (z, δ), F (z, δ)) = N(z), where

N(z) = −√−1

2

∫

C

η ∧ η = −∫

C

|z0 − ζ|−2µ0 · · · |zn − ζ|−2µn |d(area).

Proof. The assertions about the signature ofH involve a linear algebra calculation thatwe leave to the reader (who may consult [4]). We do the hyperbolic case first, so assume1 < |µ| < 2. First notice that the integral definingN(z) converges (here we use that|µ| > 1) and takes on a value which is real and negative. We claim that

(1.4) N(z) =∑

1≤j<k≤n+1

wjwkFj(z, δ)Fk(z, δ).

To see this, let us integrateη = ηz, using the determination defined byδ: Φ(ζ) :=∫ ζ

z0η,

where the path of integration is not allowed to crosssupp(δ). We havedΦ = η outsidesupp(δ) and by Stokes theorem

N(z) = −√−1

2

∫

C

η ∧ η =

√−1

2

∫

C

d(Φη) =

√−1

2

n+1∑

k=1

(∫

δk

Φη −∫

δ−k

Φη

).

As to the last terms, we observe that onδk we haveΦ(ζ) =∑

j<k wjFj +∫ ζ

zk−1η (we

abbreviateFj(z, δ) by Fj), where the last integral is taken over a subarc ofδk. Likewise,

onδ−k : (Φ|δ−k )(ζ) =∑

j<k wjFj +∫ ζ

zk−1w2

kη. Hence onδk we have

Φη − (Φη|δ−k ) =∑

j<k

(wj Fj +

∫ ζ

zk−1

η

)η −

∑

j<k

(wjFj +

∫ ζ

zk−1

w2k η

)w2

kη =

=∑

j<k

(wj − wjw

2k

)Fjη =

∑

j<k

(wjwk − wjwk) Fjwkη,

which after integration overδk yields∫

δk

Φη −∫

δ−k

Φη =∑

j<k

(wjwk − wjwk) FjFk =2√−1

∑

j<k

Im(wjwk)FjFk.

Our claim follows if we substitute this identity in the formula forN above.We continue the proof. The claim implies thatH(F (z, δ), F (z, δ)) = N(z). The

functionN is obviouslyΓ-invariant (it does not involveδ). SinceN determinesH , so isH . So this settles the hyperbolic case.

For the elliptic and parabolic cases we may verify by hand that it is invariant undera generating set of monodromy transformations, but a computation free argument, basedanalytic continuation as in [4], is perhaps more satisfying. It runs as follows: if we choosea finite set of generatorsα1, . . . , αN of PBrn+1, then for every weight systemµ we havea projective linear transformationPρµ(αi) of Pn−1 that depends in a real-analytic manneron µ. We will see that the Hermitian formshµ defined on an open subset of the tangentbundle ofPn−1 also depend real-analytically onµ; so if hµ is preserved by thePρµ(αi)’s

12 EDUARD LOOIJENGA

for a nonempty open subset ofµ’s, then it is preserved for all weight systems for whichthis makes sense. HencePρµ(αi) multipliesH by a scalar. For1 < |µ| < 2 this scalar isconstant 1. Another analytic continuation argument implies that it is 1 for allµ.

1.7. Cohomological interpretation via local systems of rank one. We sketch a settingin terms of which the Hermitian formH is best understood. It will not play a role in whatfollows (hence may be skipped), although it will reappear ina more conventional context(and formally independent of this discussion) in Section 4.The reader should consult§2of [5] for a more thorough treatment.

Fix complex numbersα0, . . . , αn in C×. LetL be a local system of rank one onU :=C − z0, . . . , zn = P1 − z0, . . . , zn+1 such that the (counterclockwise) monodromyaroundzk is multiplication byαk. It is unique up to isomorphism. We fix a nonzeromultivalued sectione of L by choosing a nonzero section ofL on some left half planeand then extend that section to the universal cover ofU (defined by that left half plane).Denote byL := OU ⊗C L the underlying holomorphic line bundle. So ifµk ∈ C is suchthatexp(2πµk

√−1) = αk, thens(ζ) :=

∏nk=1(zk − ζ)−µk ⊗ e can be understood as a

generating section ofL. Likewise,sdζ is a generating ofΩ(L) = ΩU ⊗C L. Notice thatLcomes with a connection∇ : L → Ω(L) characterized by

∇(s) =

(n∑

k=0

µk

zk − ζ

)sdζ

and thatL is recovered from the pair(L,∇) as the kernel of∇.The topological Euler characteristic of a rank one local system on a space homotopy

equivalent to a finite cell complex is independent of that local system and hence equal tothe topological Euler characteristic of that space. So the topological Euler characteristicof L is −n. Now assume thatαk 6= 1 for all k. This ensures thatL has no nonzerosection. As there is no cohomology in degrees6= 0, 1, this implies thatdimH1(L) = n.Moreover, if j : U ⊂ P1 is the inclusion, then the stalk ofj∗L in zk is represented bythe sections ofL on a punctured neighborhood ofzk, hence is zero unlessk = n+ 1 andα0 · · ·αn = 1: then it is nonzero. So the map of complexesj!L → j∗L has cokernel aone-dimensional skyscraper sheaf at∞ or is an isomorphism. This implies that for thenatural mapi : H1

c (L) → H1(L), dimKer(i) = dimCoker(i) is 1 or 0, depending onwhether or notα0 · · ·αn = 1. We denote the image ofi by IH1(L).

A relative arcα plus a section ofL∨ over its relative interior defines a relative cycleof (P1, z0, . . . , zn+1) with values inL∨ and hence an element[α] of the relative homol-ogy spaceH1(P1, z0, . . . , zn+1;L∨). Alexander duality identifies the latter cohomologyspace with the dual ofH1(L). To make the connection with the preceding, let us identifyη with sdζ (we need not assume here thatµk ∈ (0, 1)), so that we have a De Rham class[η] ∈ H1(L). If we are given anL-arc systemδ and choose the determination ofe on δkprescribed by the arc system, thenwk[δk]nk=1 is a basis ofH1(P1, z0, . . . , zn+1;L∨)and the value of[η] on wk[δk] is justFk(z, δ).

We have a perfect (Poincare) dualityH1c (L)×H1(L∨) → C, which, if cohomology is

represented by means of forms, is given by integration overU of the cup product. Supposenow in addition that|αk| = 1 for all k. This implies thatL carries a flat metric; wechoose this metric to be the one for whiche has unit length. The metric may be viewedas aC-linear isomorphism of sheavesL → L∨ (hereL stands for the local systemL withits complex conjugate complex structure) so that our perfect duality becomes a bilinearmapH1

c (L) × H1(L) → C. We multiply that map by12√−1 and denote the resulting


sesquilinear maph : H1c (L) × H1(L) → C. Thenh is Hermitian in the sense that if

α, β ∈ H1c (L), thenh(α, i∗β) = h(β, i∗α), in particular, it induces a nondegenerate

Hermitian form onIH1(L). This is just minus the form we defined in Subsection 1.6. Ifwe takeµk ∈ (0, 1) for k = 0, . . . , n and assume1 < |µ| < 2 (so thatµn+1 ∈ (0, 1) alsoandi is an isomorphism), thenh([η], [η]) equals12

√−1∫Cη ∧ η indeed and hence equals

−N(z) = −H(F (z), F (z)).

2. ORBIFOLDS AND DISCRETE MONODROMY GROUPS

2.1. Monodromy defined by a simple Dehn twist.Let be given a relative arcγ0 in(C, z0, . . . , zn) which connectszk with zl, k 6= l. This defines a Dehn twistD(γ0)and hence an elementT of PBrn+1. We determine the action ofT on Cn. For this weneed to makeηz univalued. Suppose we are given a straight piece of arcγ1 that begins inzl, but is otherwise disjoint fromγ0 so that a neighborhood ofsupp(γ0) minussupp(γ0γ1)is simply connected. Then choose a determination forηz on this simply connected opensubset and letηz|γ0 be the limit from the left. Letγ resp.γ′ be an arc which ends inzkresp.zl, but otherwise avoidsz0, . . . , zn ∪ supp(γ0γ1) (we also assume thatγ′ stays onthe right ofγ1γ0). Then from a picture one sees that

∫

T (γ)

ηz =

∫

γ

ηz + (1 − w2l )

∫

γ0

ηz,

∫

T (γ)

ηz =

∫

γ

ηz + (−w2l + w2

kw2l )

∫

γ0

ηz ,

∫

T (γ0)

ηz = w2kw

2l

∫

γ0

ηz .

Remembering thatw2kw

2l = e2π

√−1(µk+µl), one easily deduces from these formulae:

Corollary 2.1. If µk + µl 6= 1, thenT acts inCn semisimply as a complex reflectionover an angle2π(µk + µl). If µk + µl = 1, thenT acts inCn as a nontrivial unipotenttransformation. In particular,T acts with finite order if and only ifµk + µl is a rationalnumber6= 1.

By a complex reflectionwe mean here a semisimple transformation which fixes a hy-perplane pointwise. In the elliptic and hyperbolic cases,T will be an orthogonal reflectionwith respect the Hermitian formH ; in the parabolic case, it will be restrict toAn−1 as anorthogonal affine reflection.

2.2. Extension of the evaluation map.TheΓ-coveringV n → V

n can sometimes be ex-tended as a ramifiedΓ-covering over a bigger open subsetV f

n ⊃ V n of Vn (the superscript

f stands forf inite ramification; we may writeV f(µ)n instead ofV f

n if such precision is

warranted). This means that we find a normal analytic varietyV fn which containsV

n asan open-dense subset and to which theΓ-action extends such that theΓ-orbit space canbe identified withV f

n . This involves a standard tool in analytic geometry that presumablygoes back to Riemann and now falls under the heading ofnormalization. It goes like this.If v ∈ Vn has a connected neighborhoodUv in V

n such that one (hence every) connectedcomponent of its preimage inV

n is finite overUv ∩V n , then theΓ-covering overUv ∩V

n

extends to a ramifiedΓ-covering overUv. The property imposed onUv is equivalent tohaving finite monodromy overUv ∩ V

n . The extension is unique and so ifV fn denotes the

set ofv ∈ Vn with this property, then a ramifiedΓ-coveringV fn → V f

n exists as asserted.

14 EDUARD LOOIJENGA

The naturality of the construction also ensures that theC×-action onV n (which covers the

C×-action onV n ) extends toV f

n .The spaceVn receives a natural stratification from the stratification ofCn+1 by its di-

agonals and since the topology ofV fn along strata does not change,V f

n is an open unionof strata. The codimension one strata are of the formDk,l, 0 ≤ k < l ≤ n, parameterizingthez for whichzk = zl, but no other equality among its components holds.

Lemma 2.2. The stratumDk,l lies inV fn if and only ifµk + µl is a rational number6= 1.

The Schwarz map extends across the preimage ofP(Dk,l)) holomorphically if and onlyif µk + µl < 1 and it does so as a local isomorphism if and only if1 − µk − µl is thereciprocal of a positive integer. If|µ| 6= 1, then the corresponding assertions also hold forthe Lauricella map.

Proof. In order thatDk,l ⊂ V fn , it is necessary and sufficient that we have finite mon-

odromy along a simple loop aroundDk,l. This monodromy is the image of a Dehn twistalong a circle separatingzk andzl from the other elements ofz0, . . . , zn. So the firstassertion follows from Corollary 2.1.

If γ0 connectszk with zl within the circle specified above, then∫γ0

ηz = (zk −zl)

1−µk−µl exp(holom). This is essentially a consequence of the identity∫ ε

0

t−µk(t− ε)−µldt = ε1−µk−µl

∫ 1

0

t−µk(t− 1)−µldt.

Suppose now thatµk+µl ∈ Q−1 and write1−µk−µl = p/q with p, q relatively primeintegers withq > 0. So the order of the monodromy isq and over the preimage of a pointof Dk,l, we have a coordinatezk,l with the property thatzk − zl pulls back tozqk,l. Hence∫γ0

ηz pulls back tozpk,l. In order that the Schwarz map extends over the preimage ofDk,l

holomorhically (resp. as a local isomorphism), a necessarycondition is that the Lauricellafunction

∫γ0

ηz (which after all may be taken as part of a basis of Lauricella functions) isholomorphic (resp. has a nonzero derivative everywhere). This means thatp > 0 (resp.p = 1). It is not hard to verify that this is also sufficient.

2.3. The elliptic and parabolic cases.Here the main result is:

Theorem 2.3 (Elliptic case). Suppose that|µ| < 1 and that for all0 ≤ k < l ≤ n,1 − µk − µl is the reciprocal of an integer. ThenΓ is a finite complex reflection groupin GL(n,C) (so that in particularV f

n = Vn) and F : Vn → Cn is a Γ-equivariantisomorphism which drops to an isomorphismVn → Γ\Cn.

SoP(Vn) acquires in these cases the structure of an orbifold modeledon Fubini-Studyspace. At the same time we prove a proposition that will be also useful later. Observe thatstratum ofVn is given by a partition of0, . . . , n: for z in this stratum we havezk = zl ifand only ifk andl belong to the same part. Let us say that this stratum isstable relative toµ if its associated partition has the property that every parthasµ-weight< 1. We denoteby V st

n ⊂ Vn (or V st(µ)n ⊂ Vn) the union of stable strata.

Proposition 2.4. Suppose that whenever0 ≤ k < l ≤ n are such thatµk + µl < 1,then 1 − µk − µl is the reciprocal of an integer. ThenV st

n ⊂ V fn , V st

n is a complexmanifold. The Lauricella map extends holomorphically overthis manifold and has thesame regularity properties as the map it extends: it is a local isomorphism when we arenot in the parabolic case, whereas in the parabolic case, theSchwarz map defines a localisomorphism toAn−1.


We shall need:

Lemma 2.5. Let f : X → Y be a local diffeomorphism from a manifold to a connectedRiemannian manifold. Assume thatX is complete for the induced metric. Thenf is acovering map.

Proof. We use the theorem of Hopf-Rinow which says that completeness is equivalent tothe property that every geodesic extends indefinitely as a geodesic. Lety ∈ Y . Chooseε > 0 such that theε-ball B(y, ε) is the diffeomorphic image of theε-ball in TyY underthe exponential map. It is enough to show that everyx ∈ f−1B(y, ε) has a neighborhoodwhich is mapped byf diffeomorphically ontoB(y, ε). SinceX is complete, there is a(geodesic) lift of the geodesic inB(y, ε) from f(x) to y which begins inx. Then the endpoint x0 of that lift lies in f−1y. ThenB(x0ε) containsx and maps diffeomorphicallyontoB(y, ε).

We now begin the proofs of Theorem 2.3 and Proposition 2.4. Let us writeAk for theassertion of Theorem 2.3 fork + 1 points andBk for the assertion of Proposition 2.3 forelliptic strata of codimension≤ k. Let us observe thatB1 holds: an elliptic stratum ofcodimension one is a stratum of the formDk,l satisfying the hypotheses of Lemma 2.2.We now continue with induction following the scheme below.

Proof thatAk impliesBk. Consider a stratum of codimensionk. Let us first assume thatit is irreducible in the sense that it is given by a single part. Without loss of generalitywe may then assume that it is the open-dense in the locusz0 = · · · = zk. This is thesituation we studied in Subsection 1.4 (mainly for this reason, as we can now confess).We found thatF extends to as a multivalued map defined on an open subset of theblowupBl(z0,...,zk) Vn going to the blowupBl(F1,...,Fk) C

n. On the the exceptional divisor,Fis the product of the Schwarz map forµ′ = (µ0, . . . , µk) and the Lauricella map for(|µ′|, µk+1, . . . , µn). Our hypothesisAk then implies that the projectivized monodromynear a point of the stratum is finite. Equation (1.2) shows that in the transversal direction(theε coordinate) the multivaluedness is like that of(ε)1−|µ′|. Sinceµi + µj ∈ Q for all0 ≤ i < j ≤ k and the sum of these numbers is1

2k(k+1)|µ′|, it follows that|µ′| ∈ Q. Sowe have also finite order monodromy along the exceptional divisor. This implies that wehave finite local monodromy at a point of the stratum: the stratum is elliptic. We provedin fact slightly more, namely that this local monodromy group is the one associated to theLauricella system of typeµ′. So we may then invokeAk to conclude thatV st

n is in factsmooth over this stratum.

In the general case, with a stratum corresponding to severalclusters forming, we havetopologically a product situation: the local monodromy group near a point of that stratumdecomposes as a product with each factor corresponding to a cluster being formed. It isclear that if each cluster is of elliptic type, then so is the stratum. Its preimage inV st

n willbe smooth.

The asserted regularity properties of this extension of theLauricalla map hold on codi-mension strata by Lemma 2.2. But then they hold everywhere, because the locus where ahomolomorphic map between complex manifolds of the same dimension fails to be a localisomorphism is of codimension≤ 1.

Proof thatBn−1 impliesAn. SinceBn−1 holds, it follows thatV fn containsVn − 0.

ThusPF : P(Vn) → Pn−1 is defined. The latter is aΓ-equivariant local isomorphism withΓ acting onP(Vn) with compact fundamental domain (for its orbit space is the compact

16 EDUARD LOOIJENGA

P(Vn)) and on the range as a group of isometries. This implies thatP(Vn) is complete.

According to Lemma 2.5,PF is then an isomorphism. HenceF : Vn−0 → Cn − 0is a covering projection. Since the domain of the latter is connected and the range issimply connected, this map is an isomorphism. In particular, P(Vn) is compact, so thatthe coveringP(Vn) → P(Vn) is finite. This means that the projectivization ofΓ is finite.On the other hand, theC×-action onVn − 0 needs a finite cover (of degree equal to the

denominator of1 − |µ|) to lift to Vn−0. This implies thatΓ is finite, so thatV fn = Vn.

It is now clear thatF : Vn → Cn is an isomorphism. It isΓ-equivariant and drops to anisomorphismVn → Γ\Cn of affine varieties.

In the parabolic caseP(Vn) acquires the structure of an orbifold modeled on flat space:

Corollary 2.6 (Parabolic case). Suppose that|µ| = 1 and that for all0 ≤ k < l ≤ n,1− µk − µl is the reciprocal of an integer. ThenΓ acts as a complex Bieberbach group inAn−1, V f

n = Vn − 0 andPF : P(Vn) → An−1 is aΓ-equivariant isomorphism whichdrops to an isomorphismP(Vn) → Γ\An−1.

Proof. It follows from Proposition 2.4 thatV fn containsVn − 0 so thatPF : P(Vn) →

An−1 is defined. The latter is aΓ-equivariant local isomorphism withΓ acting on theP(Vn)

with compact fundamental domain and on the range as a group ofisometries. HenceP(Vn)is complete. It the follows from Lemma 2.5 thatPF is aΓ-equivariant isomorphism. It alsofollows thatΓ acts onAn−1 discretely with compact fundamental domain. This group isgenerated by complex reflections, in particular it is a complex Bieberbach group. Clearly,PF induces an isomorphismP(Vn) ∼= Γ\An−1.

We have also partial converses of Theorem 2.3 and Corollary 2.6. They will be conse-quences of

Lemma 2.7. The Lauricella map extends holomorphically over any stablestratum con-tained inV f

n .

Proof. Let S ⊂ 0, . . . , n define an stable stratumDS (i.e.,S has at least two membersand

∑k∈S µk < 1) and assume thatDS ⊂ V f

n . If 0 ≤ k < l ≤ n is contained inS,thenµk + µl ≤ |µ| < 1 and so the associated monodromy transformationT is accordingto Corollary 2.1 a reflection over an angle2π(µk + µl). SinceDS ⊂ V f

n , we must haveµk+µl ∈ Q. Lemma 2.2 tells us thatF then extends holomorphically over the preimage ofDk,l. The usual codimension argument implies that this is then also so over the preimageof DS.

Proposition 2.8. If |µ| < 1 andΓ is finite, then the Lauricella map drops to a finite mapVn → Γ\Cn.

If |µ| = 1, n > 1 andΓ acts on the complex Euclidean spaceAn−1 as a complexBieberbach group, thenV f

n = Vn and the Schwarz map drops to a finite mapP(Vn) →Γ\An−1.

Proof. In the elliptic case, it follows from Lemma 2.7 that the mapF drops to a mapVn →Γ\Cn which exists in the complex-analytic category. The map in question is homogeneous(relative to the naturalC×-actions) and the preimage of0 is 0. Hence it must be a finitemorphism. In the parabolic case, the lemma implies that the Schwarz map determines amapP(Vn) → Γ\An−1 which lives in the complex-analytic category. This map willbefinite, because its fibers are discrete and its domain is compact.


3. THE HYPERBOLIC CASE

Throughout this section we always suppose that1 ≤ |µ| < 2.

3.1. A projective set-up. An important difference with the elliptic and the paraboliccasesis thatzn+1 = ∞ is now of the same nature as the finite singular points, since we haveµn+1 = 1 − |µ| ∈ (0, 1). This tells us that we should treat all the pointsz0, . . . , zn+1

on the same footing. In more precise terms, instead of takingzn+1 = ∞ and study thetransformation behavior of the Lauricella integrals underthe affine groupC×⋉C of C, weshould letz0, . . . , zn+1 be distinct, but otherwise arbitrary points ofP1 and let the groupPGL(2,C) take role of the affine group. This means in practice that we will sometimesallow some finitezk to coalesce withzn+1 (that is, to fly off to infinity). For this weproceed as follows. LetZ0, . . . , Zn+1 be nonzero linear forms onC2 defining distinctpointsz0, . . . , zn+1 of P1. Consider the multivalued2-form onC2 defined by

Z0(ζ)−µ0 · · ·Zn+1(ζ)

−µn+1dζ0 ∧ dζ1.

Let us see how this transforms under the groupGL(2,C). The subgroupSL(2,C) leavesdζ0 ∧ dζ1 invariant, and so it simply transforms underSL(2,C) via the latter’s diagonalaction on the(C2)n+2 (the space that containsZ = (Z0, . . . , Zn+1)). The subgroup ofscalars,C× ⊂ GL(2,C) leaves the2-form invariant. So the form has a pole of order oneat the projective lineP1 at infinity. We denote the residue of that form onP1 by ηZ . It isnow clear, that a Lauricella function

∫γηZ will be GL(2,C)-invariant. Since the2-form

(and henceηZ ) is homogeneous of degree−µk in Zk, it follows that the quotient of twoLauricella functions will only depend on theGL(2,C)-orbit of (z0, . . . , zn+1).

LetQµ denote theSL(2,C)-orbit space of the subset of(P1)n+2 parameterizing distinct

(n + 2)-tuples inP1. This is in a natural way a smooth algebraic variety which canbeidentified withP(V

n ) (every orbit is represented by an(n+2)-tuple of which the last pointis ∞). So we have aΓ-coveringQ

µ → Qµ and a local isomorphismPF : Q

µ → Bn−1.Thus far our treatment ofzn+1 as one of the otherzi’s has not accomplished anything, butit will matter when we seek to extend it as a ramified covering.

We say thatz = (z0, . . . , zn+1) ∈ (P1)n+2 is µ-stableresp.µ-semistableif the R-divisor Div(z) :=

∑n+1k=0 µk(zk) has no point of weight≥ 1 resp.> 1. Let us denote

the corresponding (Zariski open) subsets of(P1)n+2 by U stµ resp.U sst

µ . Notice that when

z is µ-stable, the support of∑n+1

k=0 µk(zk) has at least three points. This implies that theSL(2,C)-orbit space (denotedQst

µ ) of U stµ is in a natural manner a nonsingular algebraic

variety: given aµ-stable pointz, we can always pick three pairwise distinct componentsfor use as an affine coordinate forP1. By means of this coordinate we get a nonemptyZariski-open subset in(P1)n−1 which maps bijectively to an open subset ofQst

µ . Thesebijections define an atlas for the claimed structure. In the semistable case, we can choosea coordinate forP1 such that∞ has weight1.

Geometric Invariant Theory tells us that in case theµk ’s are all rational, one can com-pactifyQst

µ to a projective variety by adding just finitely many points: one point for eachorbit containing a point whose associated divisor is(0) + (∞) or equivalently, for eachsplitting of 0, . . . , n + 1 into two subsets, each of which of totalµ-weight1. (So if nosuch splitting exists, thenQst

µ is already projective variety.) Let us denote that projectivecompactification byQsst

µ . This is in fact a quotient of aU sstµ with the property that each

fiber is the closure of aSL(2,C)-orbit and contains a unique closedSL(2,C)-orbit (in thestrictly stable case the latter is represented by az whose divisor is(0) + (∞)).

18 EDUARD LOOIJENGA

Theorem 3.1. Assume that for every pair0 ≤ k < l ≤ n + 1 for whichµk + µl < 1,1 − µk − µl is the reciprocal of an integer. Then the monodromy coveringQ

µ → Qµ

extends to a ramified coveringQstµ → Qst

µ andF extends to aΓ-equivariant isomorphism

Qstµ → Bn−1. MoreoverΓ acts inBm discretely and with finite covolume; this action is

with compact fundamental domain if and only no subsequence of µ has weight1.

Remarks3.2. Our hypotheses imply that theµk ’s are all rational so that the GIT com-pactificationQsst

µ makes sense. The compactication ofΓ\Bn−1 that results byΓ\Bn−1∼=

Qstµ ⊂ Qsst

µ coincides with theBaily-Borel compactificationof Γ\Bn−1.The cohomology and intersection homology of the varietyQsst

µ has been investigatedby Kirwan-Lee-Weintraub [8].

Before we begin the proof of Theorem 3.1 we need to know a little bit about the be-havior of the complex hyperbolic metric on a complex ball near a cusp. LetW be afinite dimensional complex vector space equipped with a nondegenerate Hermitian formH of hyperbolic signature so thatH(w,w) > 0 defines a complex ballB(W ) ⊂ P(W ).Let e ∈ W be a nonzero isotropic vector. Since its orthogonal complement is negativesemidefinite, every positive definite line will meet the affine hyperplane inW defined byH(w, e) = −1. In this way we find an open subsetΩe in this hyperplane which maps iso-morphically ontoB(W ). This is what is called a realization ofB(W ) as a Siegel domainof the second kind.

Lemma 3.3. The subsetΩ of the affine spaceH(w, e) = −1 defined byH(w,w) <0 is invariant under translation byR≥0e. If K ⊂ Ω is compact and measurable, thenK +

√−1R≥0e is as asubset ofΩ complete and of finite volume.

Proof. This is well-known, but we outline the proof anyway. Writee0 for e and lete1 ∈ Wbe another isotropic vector such thatH(e0, e1) = 1 and denote byW ′ the orthogonalcomplement of the span ofe0, e1. So if we writew = w0e0 + w1e1 + w′ with w′ ∈ W ′,thenΩ is defined byw1 = −1 andRe(w0) >

12H(w′, w′). This shows in particular that

Ω is invariant under translation byτe, whenRe(τ) ≥ 0. Let Ko ⊂ Ω be compact balland suppose thatw ∈ Ko 7→ H(w, e1) is constant. IfR > 0, then the map(w, y, x) ∈Ko×[−R,R]×R≥0 → (w+(x+

√−1y)e0 ∈ Ω, is an embedding. It is straightforward to

verify that the pull-back of the metric ofΩ ∼= B(W ) is comparible to the ‘warped metric’x−1(gΩ|Ko

) + x−2(dx2 + dy2). From this it easily follows thatKo × [−R,R]× R≥0 iscomplete and of finite volume. Since any compact measurableK ⊂ Ω is covered by theimage of finitely many mapsKo× [−R,R]×R≥0 → Ω as above, the lemma follows.

It follows from Proposition 2.4 thatQstµ ⊂ Qf

µ and that the Schwarz mapPF : Qstµ →

Bn−1 is a local isomorphism. SoQstµ inherits a metric fromBn−1. We need to show that

Qstµ is complete and has finite volume. The crucial step toward this is:

Lemma 3.4. Let 0 < r < n be such thatµ0 + · · · + µr = 1. Denote byD the set of(z0, . . . , zn) ∈ Cn+1 satisfying|z0| < · · · < |zr| < 1 < 2 < |zr+1| < · · · < |zn| andz0+ · · ·+zr = 0. ThenD embeds inQ

µ and its closure inQstµ is complete and of bounded

volume.

Proof. ThatD embeds inQµ is clear. LetD′ ⊂ D be the open-dense subset ofz ∈ D

for which | arg(zk)| < π for all k. There is a natural isotopy class ofL-arc systemsδ foreveryz ∈ D′ characterized by the property thatδk never crosses the negative real axis and|δk| is monotonous. This defines a liftD′ of D′ to Qst

µ so that is definedF : D → Cn.


For t > 0, denote byD′(t) the set ofz ∈ D for which |z0 · · · zr|/|zr+1 · · · zn| ≥ t. It iseasy to see thatD′(t) has compact closure inQst

µ and so the closure of its preimageD′(t)

in Qstµ is compact as well.

Sinceµ0 + · · · + µr = 1, Lemma 1.9 will apply here. As in that lemma, we putF := z

−µr+1

r+1 · · · z−µn

n F . According to that lemma we have∑r

k=1 Im(wk)Fk(z) = 0.

This amounts to saying thatH(F , er+1) = −π, whereer+1 denotes the(r + 1)th basisvector ofCn. (For H(F,G) =

∑1≤j<k≤n+1 Im(wjwk)GjFk and soH(er+1, G) =∑

1≤j≤r Im(wj)Gj .) We also notice thatH(er+1, er+1) = 0. So F maps to the SiegeldomainΩ defined in Lemma 3.4 if we takee := π−1er+1. Hence the lemma will follow ifwe show that the image ofD in Ω is contained in a subset of the formK+R>0er+1. Nownotice that for0 < ε < 1, z 7→ zε mapsD(t) ontoD′(tεn+1). From Lemma 1.9 we seethat the coordinatesFk stay bounded onD′ for all k 6= r+1, whereasRe Fr+1|D′(t) → ∞ast → 0. This means thatD′ in Ω is contained in a subset of the formK +R>0er+1.

Proof of Theorem 3.1.The GIT compactificationQsstµ of Qst

µ adds a point for every per-mutationσ of 0, . . . , n for whichµσ(0) + · · · + µσ(r) = 1 for some0 < r < n. If σ issuch a permutation, then we have defined an open subsetDσ ⊂ Q

µ as in Lemma 3.4 andaccording to that Lemma, the closure ofDσ in Qst

µ is complete and of finite volume. Thecomplement inQst

µ of the union of these closures is easily seen to be compact. HenceQstµ

is complete and of finite volume. The theorem now follows fromLemma 2.5 (bearing inmind thatQsst

µ = Qstµ if and only if no subsequence ofµ has weight1).

3.2. Extending the range of applicability. We begin with stating a partial converse toTheorem 3.1, the hyperbolic counterpart of Proposition 2.8:

Proposition 3.5. Suppose that1 < |µ| < 2, n > 1 andΓ acts onBn−1 as adiscretegroup. ThenΓ has finite covolume and the Schwarz map drops to a finite morphismQst

µ →Γ\Bn−1.

Proof. It follows from Lemma 2.7 that the Schwarz map is defined overQstµ and hence

drops to a mapQstµ → Γ\Bn−1. It follows from Lemma 3.4 (by argueing as in the proof of

Theorem 3.1) thatQstµ is complete as a metric orbifold and of finite volume. This implies

thatQstµ → Γ\Bn−1 is a finite morphism.

This immediately raises the question which weight systemsµ satisfy the hypotheses ofProposition 3.5. The first step toward the answer was taken byMostow himself [9], whoobserved that if some of the weightsµk coincide, then the conditions of (2.3), (2.6) and(3.1) may be relaxed, while still ensuring thatΓ is a discrete subgroup of the relevant Liegroup. The idea is this: ifSµ denotes the group of permutations of0, . . . , n+ 1 whichpreserve the weights, then we should regard the Lauricella mapF as being multivaluedon Sµ\V

n , rather than onV n . This can make a difference, for the monodromy cover of

Sµ\V n need not factor throughV

n . We get the following variant of Lemma 2.2

Lemma 3.6. Suppose that in Lemma 2.2 we haveµk = µl ∈ Q− 12. Then the Lauricella

map (the Schwarz map if|µ| = 1) extends over the image inDk,l in Sµ\V n as a local

isomorphism if and only if12 − µk is the reciprocal of a positive integer.

Definition 3.7. We say thatµ satisfies thehalf integrality conditionsif whenever for0 ≤k < l ≤ n+1 we haveµk+µl < 1, then(1−µk−µl)

−1 is an integer or in caseµk = µl,just half an integer.

20 EDUARD LOOIJENGA

This notion is a priori weaker than Mostow’sΣINT condition, but in the end it appar-ently leads to the same set of weight systems. Now Proposition 2.4 takes the followingform.

Proposition 3.8. If µ satisfies the half integrality conditions, thenV stn ⊂ V f

n , Sµ\V stn is

nonsingular, and the Lauricella map extends holomorphically to Sµ\V stn . This extension

has the same regularity properties as the map it extends: it is a local isomorphism whenwe are not in the parabolic case, whereas in the parabolic case, the Schwarz map definesa local isomorphism toAn−1.

This leads to (see [9] and for the present version, [4]):

Theorem 3.9. Suppose thatµ satisfies the half integrality conditions.

ell: If |µ| < 1, thenΓ is a finite complex reflection group inGL(n,C) and F :

Sµ\V n → Cn is a Γ-equivariant isomorphism which drops to an isomorphismSµ\Vn → Γ\Cn.

par: If |µ| = 1, thenΓ acts as a complex Bieberbach group inAn−1, V fn = Vn − 0

andPF : P(Sµ\V n) → An−1 is aΓ-equivariant isomorphism which drops to anisomorphismP(Sµ\Vn) → Γ\An−1.

hyp: If1 < |µ| < 2, then the monodromy coveringSµ\Qµ → Sµ\Q

µ extends to a ram-

ified coveringSµ\Qstµ → Sµ\Qst

µ andF extends to aΓ-equivariant isomorphism

Sµ\Qstµ → Bn−1. MoreoverΓ acts discretely inBm and with finite covolume.

Example.Let us taken ≤ 10 andµk = 16 for k = 0, . . . , n. So we haveµn+1 = 11−n

6 .The half integrality conditions are fulfilled for alln ≤ 10 with 1 ≤ n ≤ 4, n = 5,6 ≤ n ≤ 11 yielding an elliptic, parabolic and hyperbolic case, respectively andSµ is thepermutation group of0, . . . , n for n ≤ 9 and the one of0, . . . , 11 for n = 10.

Mostow subsequently showed that in the hyperbolic range with n ≥ 3 we thus find allbut ten of the discrete monodromy groups of finite covolume: one is missed forn = 4(namely( 1

12 ,312 ,

512 ,

512 ,

512 ,

512 )) and nine forn = 3 (see [10], (5.1)). He conjectured that

in these nine casesΓ is always commensurable with a group obtained from Theorem 3.9.This was proved by his student Sauter [11]. It is perhaps no surprise that things are a bitdifferent whenn = 2 (so that we are dealing with discrete groups of automorphismof theunit disk): indeed, the exceptions then make up a number of infinite series ([10], Theorem3.8). It turns out that forn > 10 the monodromy group is never discrete and that forn = 10 this happens only whenµk = 1

6 for k = 0, . . . , 10. (It is not known whetherthere exist discrete subgroups of isometry groups of finite covolume of a complex ball ofdimension≥ 10.)

4. MODULAR INTERPRETATION

We assume here that we are in theQ-hyperbolic case:µk ∈ (0, 1) and rational fork = 0, . . . , n+ 1 (with

∑n+1k=0 µk = 2 as always).

4.1. Cyclic covers ofP1. We will show that the Schwarz map can be interpreted as a‘fractional period’ map. This comes about by passing to a cyclic cover ofP1 on which theLauricella integrand becomes a regular differential. Concretely, writeµk = dk/m withdk,m positive integers such that thedk ’s have no common divisor, and writemk for thedenominator ofµk. Consider the cyclic coverC → P1 of orderm which has ramification


overzk of ordermk. In affine coordinates,C is given as the normalization of the curvedefined by

wm =

n∏

k=0

(zk − ζ)dk .

This is a cyclic covering which has the groupGm of mth roots of unity as its Galois group:g∗(w, z) = (χ(g)w, z), whereχ : Gm ⊂ C× stands for the tautological character. TheLauricella integrand pulls back to a univalued differential η onC, represented byw−1dζso thatg∗(η) = χ(g)η. Hence, if we letGm act on forms in the usual manner (g ∈ Gm

acts as(g−1)∗), then η is an eigenvector with characterχ. It is easily checked thatη isregular everywhere.

In order to put this in a period setting, we recall some generalities concerning the Hodgedecomposition ofC: its space of holomorphic differentials,Ω(C), has dimension equal tothe genusg of C andH1(C;C) is canonically represented on the form level by the directsumΩ(C) ⊕ Ω(C) (complex conjugation on forms corresponds to complex conjugationin H1(C;C) with respect toH1(C;R)). The intersection product onH1(C;Z) definedby (α, β) 7→ (α ∪ β)[C] (where the fundamental class[C] ∈ H2(C,Z) is specified by thecomplex orientation ofC), is on the form level given by

∫C α∧β. The associated Hermitian

form onH1(C;C) defined byh(α, β) :=√−12 (α ∪ β)[C] =

√−12

∫Cα ∧ β has signature

(g, g). The Hodge decompositionH1(C;R) = Ω(C)⊕Ω(C) ish-orthogonal with the firstsummand positive definite and the second negative definite. The Hodge decomposition, theintersection product and (hence) the Hermitian formh are all left invariant by the action ofGm.

Proposition 4.1. The eigenspaceΩ(C)χ is of dimension one and spanned byη and theeigenspaceΩ(C)χ is of dimensionn−1. The eigenspaceH1(C,C)χ has signature(1, n−1) and we haveh(η, η) = −mN(F (z), F (z)).

Lemma 4.2. Let r ∈ 0, 1, . . . ,m − 1. Then the eigenspaceΩ(C)χr

is spanned by theformsw−rf(ζ)dζ wheref runs over the polynomials of degree< −1 + r

∑nk=0 µk that

have inzk a zero of order≥ [rµk], k = 0, . . . , n. In particular,dimΩ(C)χr

is the largestinteger smaller than

∑nk=0rµk (recall thata := a− [a]).

Proof. Any meromorphic differential onC which transforms according to the characterχr, r = 0, 1, . . . ,m− 1, is of the formw−rf(ζ)dζ with f meromorphic. A local compu-tation shows that in order that such a differential be regular, it is necessary and sufficientthat f be a polynomial of degree< −1 + r

∑nk=0 µk which has inzk a zero of order

> −1 + rµk, that is, of order≥ [rµk].

Proof of Proposition 4.1.If we apply Lemma 4.2 to the caser = 1, then we find thatfmust have degree< −1 +

∑nk=0 µk = 1 − µn+1 and asµn+1 ∈ (0, 1), this means thatf

is constant. Soη spansΩ(C)χ.Forr = m− 1, we find thatdimΩ(C)χ is the largest integer smaller than

∑nk=0(m−

1)µk =∑n

k=0dk −µk =∑n

k=0(1−µk) = n− 1+µn+1, that is,n− 1. SinceΩ(C)χ

is the complex conjugate ofΩ(C)χ, it follows that this space has dimensionn− 1 also.ThatH1(C,C)χ has signature(1, n − 1) is now a consequence of its orthogonal de-

composition intoΩ(C)χ andΩ(C)χ. Finally,

h(η, η) =

√−1

2

∫

C

η ∧ η =m√−1

2

∫

C

η ∧ η = −mN(z, z)(> 0).

22 EDUARD LOOIJENGA

So the Schwarz mapPF : Qstµ → Bn−1 can now be understood as attaching to the

curveC with itsGm-action the Hodge decomposition ofH1(C;C)χ.

4.2. Arithmeticity. The above computation leads to the following arithmeticitycriterionfor Γ:

Theorem 4.3.The monodromy groupΓ is arithmetic if and only if for everyr ∈ (Z/m)×−±1 we have

∑nk=0rµk ≤ 1 or

∑nk=0−rµk ≤ 1.

We need the following density lemma.

Lemma 4.4. The Zariski closure ofΓ in GL(H1(C,C)χ ⊕H1(C,C)χ) is defined overRand the image of its group of real points in the general lineargroup ofH1(C,C)χ containsthe special unitary group ofH1(C,C)χ.

The proof amounts to exhibiting sufficiently many complex reflections inΓ. It is some-what technical and we therefore omit it.

Proof of Theorem 4.3.Let us abbreviateH1(C,C)χr

by Hr. The smallest subspace ofH1(C,C) which containsH1 and is defined overQ is the sum of the eigenspacesH :=⊕r∈(Z/m)×Hr. We may identifyH with the quotient ofH1(C,C) by the span of theimages of the mapsH1(Gk\C,C) → H1(C,C), wherek runs over the divisors6= 1 of m.In particular,H(Z) := H1(C,Z)∩H spansH . The monodromy groupΓ may be regardedas a subgroup ofGL(HZ). On the other hand,Γ preserves each summandHr. So if wedenote byG theQ-Zariski closure ofΓ in GL(H), thenΓ ⊂ G(Z) andG(C) decomposesasG(C) =∏r∈(Z/m)× Gr(C) with Gr(C) ⊂ GL(Hr). To say thatΓ is arithmetic is to saythatΓ is of finite index inG(Z).

SinceHr ⊕H−r is defined overR, so isGr,−r := Gr ×G−r. According to Lemma 4.4,the image ofG1,−1(R) in Gr(C) contains the special unitary group ofH1. The summandHr with its Hermitian form is forr ∈ (Z/m)× a Galois conjugate ofH1 and so it thenfollows that the image ofGr,−r(R) in Gr(C) contains the special unitary group ofHr.

Suppose now thatΓ is arithmetic. The projectionG(R) → G1,−1(R) is injective onΓand so the kernel of this projection must be anisotropic:Gr,−r(R) is compact forr 6= ±1.This means that the Hermitian form onHr is definite forr 6= ±1. SinceHr = Ω(C)χ

r ⊕Ω(C)χ−r with the first summand positive and the second summand negative, this meansthat for everyr ∈ (Z/m)× − ±1 (at least) one of the two summands must be trivial.Following Lemma 4.2 this amounts to

∑nk=0rµk < 1 or

∑nk=0−rµk < 1.

Suppose conversely, that for all allr ∈ (Z/m)× − ±1 we have∑n

k=0rµk < 1 or∑nk=0−rµk < 1. As we have just seen, this amounts toGr,−r(R) being compact for

all r ∈ (Z/m)× − ±1. In other words, the projection,G(R) → G1,−1(R) has compactkernel. SinceG(Z) is discrete inG(R), it follows that its image inG1,−1(R) is discrete aswell. In particular,Γ is discrete inGL(H1). Following Proposition 3.5 this implies thatΓ has finite covolume inG1,−1(R). Hence it has also in finite covolume inG(R). Thisimplies thatΓ has finite index inG(Z).

Example.The case for whichn = 3, (µ0, µ1, µ2, µ3) = ( 312 ,

312 ,

312 ,

712 ) (so thatµ4 = 8

12 )satisfies the hypotheses of Theorem 3.1, hence yields a monodromy group which operatesonB2 discretely with compact fundamental domain. But the group is not arithmetic sincewe have both

∑3k=05µk = 5

3 > 1 and∑3

k=0−5µk = 73 > 1.


4.3. Working over a ring of cyclotomic integers. If we are given anL-arc systemδ,thenC → P1 comes with a section (continuous outsideδ) in much the same way as wefound a determination ofηz: for ζ in a left half plane,

∏nk=0(zk − ζ)dk has argument

< π/2 in absolute value and so has there a naturalmth root (with argument< π/2m inabsolute value); the resulting section we find there is then extended in the obvious way. Weidentify δk with its image inC under the section and thus regard it as a chain onC. Fork = 1, . . . , n, we introduce aZ[ζm]-valued1-chain onC:

εk := wk

∑

g∈Gm

χ(g)g∗δk.

Notice that the coefficientwk is anmth root of unity and so a unit ofZ[ζm]. We put it in,in order to maintain the connection with the Lauricella map.It will also have the effect ofkeeping some of the formulae simple.

Lemma 4.5. The elementεk is a 1-cycle onC with values inZ[ζm] and has the prop-erty that g∗εk = χ(g)εk (and hence defines an element ofH1(C,Z[ζm])χ). We have∫εk

η = mFk(z, δ). Moreover,H1(C,Z[ζm])χ is as aZ[ζm]-module freely generated byε1, . . . , εn.

Proof. The identity involving integrals is verified by∫

εk

η = wk

∑

g∈Gm

χ(g)

∫

g∗δk

η = wk

∑

g∈Gm

χ(g)

∫

δk

g∗η =

= wk

∑

g∈Gm

χ(g)

∫

δk

χ(g)η = mwk

∫

δk

η = mFk(z, δ).

GiveP1 the structure of a finite cell complex by taking the singletons z0, . . . , zn as0-cells, the intervalsδ1, . . . , δn minus their end points as1-cells andP1 − ∪n

i=kδk as2-cell.The connected components of the preimages of cells inC give the latter the structure of afinite cell complex as well (over the2-cell we have one point of ramification, namely∞,and so connected components of its preimage are indeed2-cells). The resulting cellularchain complex ofC,

0 → C2 → C1 → C0 → 0,

comes with aGm-action. Notice thatC1 is the freeZ[Gm]-module generated byδ1, . . . , δn.On the other hand,C0

∼= ⊕nk=0Z[Gm/Gmk

] andC2∼= Z[Gm/Gmn+1

], so that(C0)χ =

(C2)χ = 0. The remaining assertions of the lemma follows from this.

We explicitly describe the Hermitian form on the freeZ[ζm]-moduleH1(C,Z[ζm])χ:

Proposition 4.6. The Hermitian formH = − 1mh is on the basis(ε1, . . . , εn) given as

follows: for1 ≤ l ≤ k ≤ n we have

H(εk, εl) =

0 if l < k − 1,

− 14 sin(π/m)−1 if l = k − 1,

14 (cot(π/mk−1) + cot(π/mk)) if l = k.

It is perhaps noteworthy that this proposition shows that the matrix ofH on ε1, . . . , εnonly involves the denominators of the weigthsµ0, . . . , µn. The proof relies on a localcomputation of intersection multiplicities with values inZ[ζm]. The basic situation is thefollowing. Consider theGm-coveringX over the complex unit disk∆ defined bywm =zd, whered ∈ 1, . . . ,m−1 andg ∈ Gm acts asg∗w = χ(g)w. The normalizationX of

24 EDUARD LOOIJENGA

X consists ofe := gcd(d,m) copies∆, ∆kk∈Z/e, as follows: if we writem = em and

d = ed andtk is the coordinate of∆k, then∆k → X is given byz = tmk andw = ζkmtdk,so that on∆k, wm = ζkmm tdmk = ζke z

d. If g1 ∈ Gm is such thatχ(g1) = ζm, theng∗1(tk+1) = tk k = 0, 1 . . . , e− 1 andg∗1t0 = ζmte−1 (becausew|∆k+1 = ζk+1

m tdk+1 and

(g∗1w)|∆k = ζmw|∆k = ζk+1m tdk).

Chooseθ ∈ (0, 2π) and letδ resp.δ′ be the ray on∆0 defined byt0 = r resp.t0 =r exp(

√−1θ/m) with 0 ≤ r < 1. We regard either as a chain with closed support. Notice

thatz mapsδ resp.δ′ onto [0, 1) resp. a ray6= [0, 1). Consider theZ[ζm]-valued chainswith closed support

δ :=∑

g∈Gm

χ(g)g∗δ, δ′ :=∑

g∈Gm

χ(g)g∗δ′.

These are in fact1-cycles with closed support which only meet in the preimage of theorigin (a finite set). So they have a well-defined intersection number.

Lemma 4.7. We haveδ · δ′ = mζm(ζm − 1)−1 = 12m(1−

√−1 cot(π/m)).

Proof. This intersection product gets a contribution from each connected component∆k.Because of theGm-equivariance these contributions are the same and so it is enough toshow that the contribution coming from one of them is(m/2e)(1 +

√−1 cot(π/2m)) =

12m(1 +

√−1 cot(π/2m)). This means that there is no loss in generality in assuming

thatd andm are relative prime. Assuming that this is the case, then we can compute theintersection product if we writeδ and δ′ as a sum of closed1-cycles with coefficients inZ[ζm]. This is accomplished by

δ =∑

g∈Gm

χ(g)g∗δ =

=

m∑

k=1

(1 + ζm + · · ·+ ζk−1m )(gk−1

1∗ δ − gk1∗δ) =m∑

k=1

1− ζkm1− ζm

(gk−11∗ δ − gk1∗δ),

(notice thatgk−11∗ δ − gk1∗δ is closed, indeed) and likewise forδ′. We thus reduce our task

to computing the intersection numbers(gk−11∗ δ− gk1∗δ) · (gl−1

1∗ δ′ − gl1∗δ′). This is easy: we

find that this equals1 if l = k, −1 if l = k − 1 and0 otherwise. Thus

δ · δ′ =m∑

k=1

1− ζkm1− ζm

ζk−1m =

mζmζm − 1

=mζ2m

ζ2m − ζ2m= 1

2m(1−√−1 cot(π/m)).

Proof of 4.6. We may of course assume that eachzk is real:zk = xk ∈ R with with x0 <

x1 < · · · < xn and thatδk = [xk−1, xk]. Let us putδk := wkεk =∑

g∈Gmχ(g)g∗δk

and computeδk · δl for 1 ≤ l ≤ k ≤ n. It is clear that this is zero in casel < k − 1.For l = k, we let δ′k go in a straight line fromxk−1 to a point in the upper half plane(with real part12xk−1 + 1

2xk, say) and then straight toxk. We have a naturally definedZ[ζm]-valued1-chain δ′k on C homologous toδk and with support lying overδk. So

δk · δk = δk · δ′k. The latter is computed with the help of Lemma 4.7: the contributionoverxk−1 is 1

2m(1 −√−1 cot(π/mk−1)) and overxk it is − 1

2m(1 −√−1 cot(π/mk))

and soεk · εk = δk · δ′k = − 12m

√−1 cot(π/mk−1)) +

12m

√−1 cot(π/mk). We now do

the casel = k − 1. The1-chains onC given byδk−1 andδk make an angle overxk−1 of


πµk−1 = πdk−1/m. In terms of the local picture of Lemma 4.7 this means that thepair(δk, δk−1) corresponds to(δ,−ζ

dk−1−12m δ′). It follows that

δk · δk−1 = δ · −ζdk−1−12m δ′ = −ζ

dk−1−12m δ · δ′ =

= −ζdk−1−12m mζm(ζm − 1)−1 = −m(ζ2m − ζ2m)−1e

√−1πµk−1 .

Henceεk · εk−1 = −m(ζ2m − ζ2m)−1 and soH(εk, εk−1) = − 12m

√−1

εk · εk−1 =

(2√−1(ζ2m − ζ2m))−1 = − 1

4 (sin(π/m))−1 is as asserted.

5. GENERALIZATIONS AND OTHER VIEW POINTS

5.1. Higher dimensional integrals. This refers to the situation whereP1 and the subsetz0, . . . , zn+1 are replaced by a projective arrangement; such generalizations were con-sidered by Deligne, Varchenko [15] and others. To be specific, fix an integerN ≥ 1, afinite setK with at leastN + 2 elements and aweight functionµ : k ∈ K 7→ µk ∈ (0, 1).Given an injective mapz : k ∈ K 7→ zk ∈ PN , choose for everyk ∈ K a linear formZk : CN+1 → C whose zero set is the hyperplaneHzk defined byzk and put

ηz = ResPN

(∏

k∈K

Zk(ζ)−µk

)dζ0 ∧ · · · ∧ dζN .

This is a multivalued holomorphicN -form onUz := PN −∪k∈KHzk . If σ is a sufficientlyregular relativeN -chain of the pair(PN ,PN − Uz) and we are given a determination ofηoverσ, thenη is integrable overσ so that

∫ση is defined. Here it pays however to take the

more cohomological approach that we briefly described in Subsection 1.7. So we letLz

be the rank one local system onUz such that its monodromy aroundHzk is multiplicationby exp(2πµk

√−1) and endow it with a flat Hermitian metric. Then after the choice of a

multivalued section ofLz of unit norm,ηz can be interpreted as a section ofΩNUz

⊗C Lz .It thus determines an element[ηz ] ∈ HN (Lz). Similarly, σ plus the determination ofηz overσ defines an element[σ] ∈ HN (PN ,PN − Uz;L∨

z ). The latter space is dual toHN(Lz) by Alexander duality in such a manner that

∫σ ηz is the value of the Alexander

pairing on([ηz ], [σ]). In order thatηz is square integrable it is necessary and s ufficientthat for every nonempty intersectionL of hyperplanesHzk we have

∑k |Hz

k⊃L µk <

codim(L). Assume that this is the case. Thenηz defines in fact a class in the intersectionhomology spaceIHm(PN ,Lz). This space comes a natural hermitian formh for whichh(ηz, ηz) > 0. (It is clear that the line spanned byηz only dependsz; Hodge theory tellsus that the image of that line isFNIHN (PN ,L).) So in order that the situation is like theone we studied we would want that the orthogonal complement of ηz in IHN (PN ,Lz) tobe negative. Unfortunately this seems rarely to be the case whenN > 1. When that is so,then we might varyz over the connected constructible setS of injective mapsK → PN

for which the the topological type of the arrangement it defines stays constant. Then overS we have a local systemHS whose stalk atz ∈ S is IHN (PN ,Lz) and the Schwarz mapwhich assigns toz the line inHz defined byηz will take values in a ball. The first orderof business should be to determine the cases for which the associated monodromy group isdiscrete.

5.2. Geometric structures on arrangement complements.In [4] Couwenberg, Heck-man and I developed a generalization of the Deligne-Mostow theory that starts with aslightly different point of view. The point of departure is here a finite dimensional complexinner product spaceV , a finite collectionH of linear hyperplanes inV and a mapκ which

26 EDUARD LOOIJENGA

assigns to everyH ∈ H a positive real numberκH . These data define a connection∇κ onthe tangent bundle of the arrangement complementV := V − ∪h∈HH as follows. ForH ∈ H denote byπH ∈ End(V ) the orthogonal projection with kernelH and byωH thelogarithmic differential onV defined byφ−1

H dφH , whereφH is a linear form onV withkernelH . FormΩκ :=

∑H∈H κHπH ⊗ ωH and regard it as a differential onV which

takes values in the tangent bundle ofV , or rather, as a connection form on this tangentbundle: a connection is defined by

∇κ := ∇0 − Ωκ,

where∇0 stands for the usual affine connection onV restricted toV . This connection iseasily verified to be torsion free. It is well-known that sucha connection defines an affinestructure (that is, it defines an atlas of charts whose transition maps are affine-linear) pre-cisely whenthe connection is flat; the sheaf of affine-linearfunctions are then the holomor-phic functions whose differential is flat for the connection(conversely, an affine structureis always given by a flat torsion free connection on the tangent bundle). There is a simplecriterion for the flatness of∇κ in terms of linear algebra. LetL(H) denote the collectionof subspaces ofV that are intersections of members ofH and let forL ∈ L(H)HL be theset ofH ∈ H containingL. Then the following properties are equivalent:

(i) ∇ is flat,(ii) Ω ∧ Ω = 0,(iii) for every pairL,M ∈ L(H) with L ⊂ M , the endomorphisms

∑H∈HL

κHπH

and∑

H∈HMκHπH commute,

(iv) for everyL ∈ L(H) of codimension2, the sum∑

H∈HLκHπH commutes with

each of its terms.

If these mutually equivalent conditions are satisfied we call the triple (V,H, κ) a Dunklsystem.

Suppose that(V,H, κ) is such a system so thatV comes with an affine structure. IfL ∈ L(H) is irreducible (in the sense that there is no nontrivial decomposition ofHL suchthat the corresponding intersections are perpendicular),then the fact that

∑H∈HL

κHπH

commutes with each of its terms implies that this sum must be proportional to the orthogo-nal projection with kernelL, πL. A trace computation shows that the sclalar factor must beκL := codim(L)−1

∑H∈HL

κH . Let us now assume that the whole system is irreduciblein the sense that the intersection of all members ofH is reduced to the origin and that thisintersection is irreducible. We then have definedκ0 = dim(V )−1

∑H∈H κH . The con-

nection is invariant under scalar multiplication byet ∈ C× and one verifies that fort closeto 0, the corresponding affine-linear transformation is like scalar multiplication bye(1−κ0)t

if κ0 6= 1 and by a translation ifκ0 = 1. This means that ifκ0 6= 1, the affine structureon V is in fact a linear structure and that this determines a (new)projective structureonP(V ), whereas whenκ0 = 1 (theparaboliccase),P(V ) inherits an affine structurewhich makes the projectionV → P(V ) affine-linear. Notice that if(V,H, tκ) will be aDunkl system for everyt > 0. The behavior of that system (such as its monodromy) maychange dramatically if we varyt.

Before we proceed, let us show how a weight systemµ that gives rise to the Lauri-cella differential also gives rise to such an irreducible Dunkl system: we takeV = Vn =Cn+1/main diagonal,H will be the collection of diagonal hyperplanesHk,l := (zk = zl),0 ≤ k < l ≤ n, andκ(Hk,l) = µk + µl. The inner product onVn comes from the innerproduct onCn+1 for which 〈ek, el〉 = µkδk,l and is the one which makes the projectionCn+1 → Vn selfadjoint. It is an amusing exercise to verify that the connection is flat


indeed and that the space of affine-linear functions atz ∈ V n is precisely the space of so-

lutions of the system of differential equations we encountered in part (c) of Proposition 1.2.So the Schwarz map is now understood as a multivalued chart (in standard terminology, adeveloping map) for the new projective structure onP(V

n ). We also find thatκ0 = |µ|;more generally, an irreducible memberL ∈ L(H) is given by a subsetI ⊂ 0, . . . , nwith at least two elements (so thatL = L(I) is the locus where allzk, k ∈ I coincide) andκL(I) =

∑k∈I µk.

Another interesting class of examples is provided by the finite complex reflection groups:let G be a finite complex reflection group operating irreducibly and unitarily in a complexinner product spaceV , H the collection of complex hyperplanes ofG andH ∈ H 7→ κH

constant on theG-orbits. Then(V,H, κ) is a Dunkl system.It turns out that in many cases of interest (including the examples mentioned above), one

can show that there exists a∇κ-flat Hermitian formh onV with the following properties

ell: if 0 < κ0 < 1, thenh is positive definite,par: if κ0 = 1, thenh positive semidefinite with kernel the tangent spaces to the

C×-orbits,hyp: if 1 < κ0 < mhyp for somemhyp > 1, thenh is nondegenerate hyperbolic and

such that the tangent spaces to theC×-orbits are negative.

This implies thatP(V ) acquires a geometric structure which is respectively modeled onFubini-Study space, flat complex Euclidean space and complex hyperbolic space. A suit-able combination of rationality and symmetry conditions (which generalizes the half inte-grality condition 3.7) yields a generalization of Theorem 3.9. We thus obtain new examplesof groups operating discretely and with finite covolume on a complex ball.

REFERENCES

[1] W. Casselman:Families of curves and automorphic forms, Thesis, Princeton University, 1966 (unpublished).[2] P.B. Cohen, F. Hirzebruch:Review ofCommensurabilities among lattices inPU(1, n) by Deligne and

Mostow, Bull. Amer. Math. Soc. 32 (1995), 88–105.[3] W. Couwenberg:Complex Reflection Groups and Hypergeometric Functions, Thesis (123 p.), University of

Nijmegen, 1994, also available athttp://members.chello.nl/∼w.couwenberg/

[4] W. Couwenberg, G. Heckman, E. Looijenga,Geometric structures on the complement of a projective ar-rangement.To appear in Publ. Math. IHES, available atarXiv math.AG/0311404

[5] P. Deligne, G.D. Mostow:Monodromy of hypergeometric functions and non-lattice integral monodromy,Publ. Math. IHES63 (1986), 1–89.

[6] P. Deligne, G.D. Mostow:Commensurabilities among lattices inPU(1, n), Ann. of Math. Studies132,Princeton U.P., Princeton 1993.

[7] B.R. Doran:Intersection Homology, Hypergeometric Functions, and Moduli Spaces as Ball Quotients, The-sis, Princeton University (93 p.), 2003.

[8] F.C. Kirwan, R. Lee, S.H. Weintraub:Quotients of the complex ball by discrete groups, Pacific J. of Math.130(1987), 115–141.

[9] G.D. Mostow: Generalized Picard lattices arising from half-integral conditions, Inst. HautesEtudes Sci.Publ. Math.63 (1986), 91–106.

[10] G.D. Mostow:On discontinuous action of monodromy groups on the complexn-ball, J. Amer. Math. Soc.1 (1988), 555–586.

[11] J.K. Sauter, Jr.:Isomorphisms among monodromy groups and applications to lattices inPU(1, 2), PacificJ. Math.146(1990), 331–384.

[12] H.A. Schwarz:Uber diejenigen Falle in welchen die Gaussische hypergeometrische Reihe eine algebraischeFunktion ihres vierten Elementes darstellt, J. f. d. reine u. angew. Math. 75 (1873), 292–335.

[13] G. Shimura:On purely transcendental fields of automorphic functions ofseveral variables, Osaka J. Math.1 (1964), 1–14.

http://arxiv.org/abs/math/0311404

28 EDUARD LOOIJENGA

[14] W.P. Thurston:Shapes of polyhedra and triangulations of the sphere, Geometry & Topology Monographs1 (1998), 511–549.

[15] A.N. Varchenko: Hodge filtration of hypergeometric integrals associated with an affine configuration ofgeneral position and a local Torelli theorem, in: I.M. Gelfand Seminar, Adv. Soviet Math.16 Part 2, 167–177, Amer. Math. Soc., Providence, RI 1993.

BETAFACULTEIT UNIVERSITEITUTRECHT–DEPARTEMENTWISKUNDE, POSTBUS80.010, NL-3508 TAUTRECHT, NEDERLAND

E-mail address: [email protected]

UNIFORMIZATION BY LAURICELLA FUNCTIONS—AN OVERVIEW … · 2. Orbifolds and discrete monodromy...

Documents

Transcript of UNIFORMIZATION BY LAURICELLA FUNCTIONS—AN OVERVIEW … · 2. Orbifolds and discrete monodromy...