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Weil-Petersson Volumes of the Moduli Spaces of CY Manifolds

arXiv:hep-th/0408033v8 25 Mar 2007

Andrey Todorov University of California, Department of Mathematics Santa Cruz, CA 95064 Bulgarian Academy of Sciences, Institute of Mathematics ul. Acad. Georgy Bonchev No 8 So?a 1113, Bulgaria March 12, 2006

Abstract In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [11]. He proved that the Chern forms of good metrics de?ne classes of cohomology with integer coe?cients on the compacti?ed quasi-projective varieties by adding a divisor with normal crossings. Viehweg proved that the moduli space of CY manifolds is a quasi-projective variety. The proof that the volume of the moduli space of polarized CY manifolds are rational number is based on the facts that the L2 norm on the dualizing line bundle over the moduli space of polarized CY manifolds is a good metric. The Weil-Petersson metric is minus the Chern form of the L2 metric on the dualizing line bundle. This fact implies that the volumes of Weil-Petersson metric are rational numbers. Also we get that the Weil-Petersson metric is a good metric. Therefore all the Chern forms de?ne integer classes of cohomologies.

Contents

1 Introduction 1.1 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Description of the Content of the Paper . . . . . . . . . . . . . . 1.3 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 4 4

1

2 Moduli of Polarized CY Manifolds 2.1 Local Moduli . . . . . . . . . . . . . . . . . . . . 2.2 A?ne Flat coordinates in the Kuranishi Space . 2.3 Weil-Petersson Metric . . . . . . . . . . . . . . . 2.4 Global Moduli . . . . . . . . . . . . . . . . . . . 2.5 A?ne Flat Coordinates around Points at In?nity

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4 4 5 7 8 9

3 Metrics on Vector Bundles with Logarithmic Growth 13 3.1 Mumford Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Example of a Good Metric . . . . . . . . . . . . . . . . . . . . . . 15 4 Applications of Mumford Theory to the Moduli of CY 17 4.1 The L2 Metric is Good . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 The Weil-Petersson Volumes are Rational Numbers . . . . . . . . 22

1

1.1

Introduction

General Remarks

There are several metrics naturally de?ned on the moduli space of Riemann surfaces. One of them is the Weil-Petersson metric. The Weil-Petersson metric is de?ned because of the existence of a metric with a constant curvature on the Riemann surface. Its curvature properties were studied by Ahlfors, Bers, S. Wolpert and so on. The generalization of the Weil-Petersson metric on the moduli space of higher dimensional projective varieties was ?rst introduced by Y.-T. Siu. He gave explicit formulas for the curvature of Weil-Petersson metric. See [12]. The generalization is possible thanks to the solution of Calabi conjecture due to Yau. See [18]. For Calabi-Yau manifolds it was noticed in [14] and [13] that the Weil-Petersson metric can be de?ned and computed by using the cup product of (n ? 1, n) forms. Another metric naturally de?ned on the moduli space of polarized CY manifolds is the Hodge metric. The holomorphic sectional curvature of the Hodge metric is negative and bounded away from zero. The holomorphic curvature of the Weil-Petersson metric is not negative. Recently some important results about the relations between the Weil-Petersson metric and Hodge metric were obtained. See [3], [8], [9] and [10]. Ph. Candelas and G. Moore asked if the Weil-Petersson volumes are ?nite. For the importance and the physical interpretation of the ?niteness of the Weil-Petersson volumes to string theory see [2], [4] and [16]. In this paper we will answer Candelas-Moore question. Moreover we will prove that the WeilPetersson volumes are rational numbers. I was informed by Prof. Lu that he and Professor Sun also proved the rationality of the volumes. See [10]. In 1976 D. Mumford introduced the notion of good metrics on vector bundles on quasi-projective varieties in [11]. He proved that the Chern forms of good metrics de?ne classes of cohomology with integer coe?cients on the compacti?ed 2

quasi-projective varieties by adding a divisor with normal crossings. Viehweg proved that the moduli space of prioritized CY manifolds is a quasi-projective variety. The idea of this paper is to apply the results of Mumford to the moduli space of CY manifolds. We proved that the L2 metric on the dualizing sheaf is good. It was proved in [14] that the Chern form of the L2 metric on the dualizing sheaf de?nes the Weil-Petersson metric on the moduli space. See also [13]. So if we prove that the L2 metric on the dualizing sheaf is good then it will imply that the Weil-Petersson volumes are rational numbers. We will explain what is the meaning of a metric on a line bundle is good one. According to [17] the moduli space of polarized CY manifolds ML (M) is a quasi-projective variety. Let ML (M) be some projective compacti?cation of ML (M) such that D = ML (M) ? ML (M) is a divisor of normal crossings. The meaning that the metric h on a line bundle over ML (M) is good is the following; Let ¦Ó¡Þ ¡Ê D, let DN be an open polydisk containing ¦Ó¡Þ , then h is a good metric on some line bundle L de?ned on ML (M) if the curvature form of the metric of the line bundle around open sets D N ? D N ¡É D = ( D ? ) ¡Á D N ?k is bounded from above by the Poincare metric on (D? ) plus the standard metric on DN ?k . This implies that if we integrate the maximal power of the curvature form over ML (M) we get a ?nite number. Moreover such curvature forms are forms with coe?cients distribution in the sense of Schwarz and they de?ne classes of cohomology of H 2 ML (M), Z . Our proof that the L2 metric h on the relative dualizing sheaf is a good metric is based on the construction of a canonical family of holomorphic forms ¦Ø¦Ó on the Kuranishi space given in [14]. The canonical family of holomorphic forms de?nes a special holomorphic local coordinates in the Kuranishi space where the components of the Weil-Petersson metric are given by 1 gi,j = ¦Äi,j + Ri,j,k,l ¦Ó k ¦Ó l + ... 6 Since h(¦Ø¦Ó ) = ¦Ø¦Ó then around a point we can compute explicitly h. Let DN be any polydisk in ML (M) containing ¦Ó¡Þ . Let k DN ? DN ¡É D = (D? ) ¡Á DN ?k . Let ¦Ð :U1 ¡Á ... ¡Á Uk ¡ú (D? ) be the uniformization map. From the results in [14] we deduce the following explicit formula for the pull back of the L2 metric 3

k 2 k k

= (?1)

n(n?1) 2

¡Ì ? ?1

n M

¦Ø¦Ó ¡Ä ¦Ø¦Ó

¦Ó¡Þ ¡Ê D = ML (M) ? ML (M)

on the relative dualizing sheaf h on U1 ¡Á ... ¡Á Uk ¡Á DN ?k : ¦Ø¦Ó

k 2

|(D? )k ¡ÁDN ?k := h|(D? )k ¡ÁDN ?k =

N

¦Ø¦Ó

2

:= h(¦Ó, ¦Ó ) :=

i=1 i j

1 ? |¦Ó i |2 +

j =k+1

1 ? |tj |2 + ¦Õ(¦Ó, ¦Ó ) + ¦·(t, t), (1)

for 0 ¡Ü |¦Ó | < 1, 0 ¡Ü t < 1 where ¦Õ(¦Ó, ¦Ó ) and ¦·(t, t) are bounded real analytic functions on the unit disk. The expression (1) shows that the L2 metric h is a good metric. This implies that the volumes of Weil-Petersson metrics are ?nite and they are rational numbers. Moreover it implies that Weil-Petersson metric is a good metric. So the Chern forms of it de?ne classes of cohomologies in H 2k ML (M), Z according to [11].

1.2

Description of the Content of the Paper

Next we are going to describe the content of each of the Sections in this article. In Section 2 we review the basis results from [14] and in [13] about local deformation theory of CY manifolds. We also review the results of [7] about the global deformation Theory. In Section 3 we review Mumford Theory of good metrics with logarithmic growth on vector bundles over quasi-projective varieties developed in [11]. In Section 4 we prove that the L2 metric on the dualizing line bundle over the moduli space is a good metric in the sense of Mumford. This results implies that the Weil-Petersson volumes are rational numbers.

1.3

Acknowledgements

Part of this paper was ?nished during my visit to MPI Bonn. I want to thank Professor Yu. I. Manin for his help. Special thanks to Ph. Candelas and G. Moore for drawing my attention to the problem of the ?niteness of the WeilPetersson volumes. I want to thank G. Moore for useful conversations long time ago on this topic and his useful comments.

2

2.1

Moduli of Polarized CY Manifolds

Local Moduli

Let M be an even dimensional C¡Þ manifold. We will say that M has an almost complex structure if there exists a section I ¡Ê C ¡Þ (M, Hom(T ? , T ? ) such that I 2 = ?id. T is the tangent bundle and T ? is the cotangent bundle on M. This de?nition is equivalent to the following one: Let M be an even dimensional C¡Þ manifold. Suppose that there exists a global splitting of the complexi?ed cotangent bundle T ? ? C = ?1,0 ¨’ ?0,1 , where ?0,1 = ?1,0 . Then we will say that M has an almost complex structure. We will say that an almost complex 4

structure is an integrable one, if for each point x ¡ÊM there exists an open set U ?M such that we can ?nd local coordinates z 1 , .., z n , such that dz 1 , .., dz n are linearly independent in each point m ¡Ê U and they generate ?1,0 |U . De?nition 1 Let M be a complex manifold. Let ¦Õ ¡Ê ¦£(M, Hom(?1,0 , ?0,1 )), then we will call ¦Õ a Beltrami di?erential. Since ¦£(M, Hom(?1,0 , ?0,1 )) ? ¦£(M, ?0,1 ? T 1,0 ), we deduce that locally ¦Õ ¦Á ¦Â ? can be written as follows: ¦Õ|U = ¦Õ¦Á dz ? ?z ¦Â . From now on we will denote by A¦Õ the following linear operator: A¦Õ = id ¦Õ(¦Ó ) ¦Õ(¦Ó ) id .

We will consider only those Beltrami di?erentials ¦Õ such that det(A¦Õ ) = 0. The Beltrami di?erential ¦Õ de?nes an integrable complex structure on M if and only if the following equation holds: ?¦Õ = where

n ¦Í =1 1¨Q¦Á<¦Â ¨Qn n ?=1

1 [¦Õ, ¦Õ] , 2

(2)

[¦Õ, ¦Õ] |U :=

? ¦Í ¦Í ¦Õ? ¦Á ?? ¦Õ¦Â ? ¦Õ¦Â (?? ¦Õ¦Á )

dz ¡Ä dz ?

¦Á

¦Â

? dz ¦Í

(3)

(See [5].) Kuranishi proved the following Theorem: Theorem 2 Let {¦Õi } be a basis of harmonic (0, 1) forms of H1 (M, T 1,0 ) on a Hermitian manifold M. Let G be the Green operator and let ¦Õ(¦Ó 1 , .., ¦Ó N ) be de?ned as follows: ¦Õ(¦Ó ) = 1 ? ¦Õi ¦Ó i + ? G[¦Õ(¦Ó 1 , ..., ¦Ó N ), ¦Õ(¦Ó 1 , ..., ¦Ó N )]. 2 i=1

N

(4)

There exists ¦Å > 0 such that for ¦Ó = (¦Ó 1 , ..., ¦Ó N ) such that |¦Ói | < ¦Å the tensor ¦Õ(¦Ó 1 , ..., ¦Ó N ) is a global C ¡Þ section of the bundle ?(0,1) ? T 1,0 .(See [5].)

2.2

A?ne Flat coordinates in the Kuranishi Space

Based on Theorem 2, the following Theorem is proved in [14]: Theorem 3 Let M be a CY manifold and let {¦Õi } be a basis of harmonic (0, 1) forms with coe?cients in T 1,0 . Then the equation (2) has a solution in the form:

N

¦Õ(¦Ó ) =

i=1

¦Õi ¦Ó i +

|IN |¨R2

¦ÕIN ¦Ó IN =

5

1 ? ¦Õi ¦Ó i + ? G[¦Õ(¦Ó 1 , ..., ¦Ó N ), ¦Õ(¦Ó 1 , ..., ¦Ó N )] 2 i=1

?

N

(5)

and there exists ¦Å > 0 such that when |¦Ó i | < ¦Å ¦Õ(¦Ó ) ¡Ê C ¡Þ (M, ?0,1 ? T 1,0 ) where i = 1, ..., N.

and ? ¦Õ(¦Ó 1 , ..., ¦Ó N ) = 0, ¦ÕIN ¦ØM = ?¦×IN where IN = (i1 , ..., iN ) is a multiindex, ¦ÕIN ¡Ê C ¡Þ (M, ?0,1 ? T 1,0 ), ¦Ó IN = (¦Ó 1 )i1 ...(¦Ó N )iN

De?nition 4 Theorem 3 implies that the Kuranishi space K is de?ned as follows: Let ¦Å > 0 be such that the Beltarmi di?erentials ¦Õ(¦Ó ) de?ned by (5) satisfy Theorem 2, then K : {¦Ó = (¦Ó 1 , ..., ¦Ó N )||¦Ó i | < ¦Å}. Thus ¦Ó = (¦Ó 1 , ..., ¦Ó N ) such that |¦Ó i | < ¦Å is a local coordinate system in K. It will be called the ?at coordinate system in K.

It is a standard fact from Kodaira-Spencer-Kuranishi deformation theory that for each ¦Ó = (¦Ó 1 , ..., ¦Ó N ) ¡Ê K as in Theorem 3 the Beltrami di?erential ¦Õ(¦Ó 1 , ..., ¦Ó N ) de?nes a new integrable complex structure on M. This means that the points of K, where K : {¦Ó = (¦Ó 1 , ..., ¦Ó N )||¦Ó i | < ¦Å} de?nes a family of operators ? ¦Ó on the C ¡Þ family K ¡Á M ¡ú M and ? ¦Ó are integrable in the sense of Newlander-Nirenberg. Moreover it was proved by Kodaira, Spencer and Kuranishi that we get a complex analytic family of CY manifolds ¦Ð : X ¡ú K, where as C ¡Þ manifold X ? K¡ÁM. The family ¦Ð:X ¡úK is called the Kuranishi family. The operators ? ¦Ó are de?ned as follows: De?nition 5 Let {Ui } be an open covering of M, with local coordinate system k {zi } where k = 1, ..., dimC M= n. We know that the Beltrami di?erential is given by: n ? k dz j ? k . (¦Õ(¦Ó 1 , ..., ¦Ó N ))j ¦Õ(¦Ó ) = ?z

j,k=1

(6)

Then it de?nes the ?¦Õ operator associated with the new complex structure as follows: n ? k ? . (7) (¦Õ(¦Ó 1 , ..., ¦Ó N ))j ?¦Õ ¦Ó,j = ? j ?z k ?z k=1 In [14] the following Theorems were proved:

6

Theorem 6 There exists a family of holomorphic forms ¦Ø¦Ó of the Kuranishi family (6) such that in the coordinates (¦Ó 1 , ..., ¦Ó N ) we have ¦Ø¦Ó = ¦Ø0 ? (¦Ø0 ¦Õi ) ¦Ó i +

i,j i,j

¦Ø0 (¦Õi ¡Ä ¦Õk ) ¦Ó i ¦Ó j + O(3).

(8)

Theorem 7 There exists a family of holomorphic forms ¦Ø¦Ó of the Kuranishi family (6) such that in the coordinates (¦Ó 1 , ..., ¦Ó N ) we have [¦Ø¦Ó ], [¦Ø¦Ó ] = (?1) 1?

n(n?1) 2

¡Ì

?1

n M

¦Ø¦Ó ¡Ä ¦Ø¦Ó =

¦Ø0 ¦Õi , ¦Ø0 ¦Õj ¦Ó i ¦Ó j +

i,j

i,j

¦Ø0 (¦Õi ¡Ä ¦Õk ) , ¦Ø0 (¦Õj ¡Ä ¦Õl ) ¦Ó i ¦Ó j ¦Ó k ¦Ó l + O(¦Ó 5 ) = ¦Ø0 (¦Õi ¡Ä ¦Õk ) , ¦Ø0 (¦Õj ¡Ä ¦Õl ) ¦Ó i ¦Ó j ¦Ó k ¦Ó l + O(¦Ó 5 ) [¦Ø¦Ó ], [¦Ø¦Ó ] ¡Ü [¦Ø0 ], [¦Ø0 ] . (9)

1? and

¦Ó i¦Ó j +

i,j i,j

2.3

Weil-Petersson Metric

It is a well known fact from Kodaira-Spencer-Kuranishi theory that the tangent space T¦Ó,K at a point ¦Ó ¡Ê K can be identi?ed with the space of harmonic (0,1) forms with values in the holomorphic vector ?elds H1 (M¦Ó , T ). We will view (0,1) (1,0) each element ¦Õ ¡Ê H1 (M¦Ó , T ) as a point wise linear map from ?M¦Ó to ?M¦Ó . 1 Given ¦Õ1 and ¦Õ2 ¡Ê H (M¦Ó , T ), the trace of the map ¦Õ1 ? ¦Õ2 : ?M¦Ó ¡ú ?M¦Ó

n (0,1) (0,1)

at the point m ¡ÊM¦Ó with respect to the metric g is simply given by: T r(¦Õ1 ? ¦Õ2 )(m) =

k m g l,k g (¦Õ1 )l (¦Õ2 )k k,m k,l,m=1

(10)

De?nition 8 We will de?ne the Weil-Petersson metric on K via the scalar product: ¦Õ1 , ¦Õ2 =

M

T r(¦Õ1 ? ¦Õ2 )vol(g ).

(11)

A very natural construction of a coordinate system ¦Ó = (¦Ó 1 , ..., ¦Ó N ) in K is constructed in [14] such that the components gi,j of the Weil Petersson metric are given by the following formulas: 7

1 gi,j = ¦Äi,j + Ri,j,l,k ¦Ó l ¦Ó k + O(¦Ó 3 ). 6

1

Very detailed treatment of the Weil-Petersson geometry of the moduli space of polarized CY manifolds can be found in [8] and [9]. In those two papers important results are obtained.

2.4

Global Moduli

De?nition 9 We will de?ne the Teichm¡§ uller space T (M) of a CY manifold M as follows: T (M) := I (M)/Dif f0 (M), where I (M) := {all integrable complex structures on M} and Di?0 (M) is the group of di?eomorphisms isotopic to identity. The action of the group Dif f0 (M) is de?ned as follows; Let ¦Õ ¡ÊDi?0 (M) then ¦Õ acts on integrable complex structures on M by pull back, i.e. if I ¡Ê C ¡Þ (M, Hom(T (M), T (M)), then we de?ne ¦Õ(I¦Ó ) = ¦Õ? (I¦Ó ). We will call a pair (M; ¦Ã1 , ..., ¦Ãbn ) a marked CY manifold where M is a CY manifold and {¦Ã1 , ..., ¦Ãbn } is a basis of Hn (M,Z)/Tor. Remark 10 Let K be the Kuranishi space. It is easy to see that if we choose a basis of Hn (M,Z)/Tor in one of the ?bres of the Kuranishi family ¦Ð : XK ¡ú K then all the ?bres will be marked, since as a C ¡Þ manifold XK ?M¡ÁK. In [7] the following Theorem was proved: Theorem 11 There exists a family of marked polarized CY manifolds ZL ¡úT(M), (12)

which possesses the following properties: a) It is e?ectively parametrized, b) For any marked CY manifold M of ?xed topological type for which the polarization class L de?nes an imbedding into a projective space CPN , there exists an isomorphism of it (as a marked CY manifold) with a ?bre Ms of the family ZL . c) The base has dimension hn?1,1 . Corollary 12 Let Y ¡úX be any family of marked polarized CY manifolds, then there exists a unique holomorphic map ¦Õ : X ¡ú T(M) up to a biholomorphic map ¦× of M which induces the identity map on Hn (M, Z). From now on we will denote by T (M) the irreducible component of the Teichm¡§ uller space that contains our ?xed CY manifold M.

1 This coordinate system is called ?at holomorphic coordinate system. It appeared for the ?rst time in [14]. Based on the information of the author of [13], it is claimed in [1] that the ?at coordinate system was introduced in [13]. The problem of the construction of the ?at holomorphic coordinates was not addressed in [13].

8

De?nition 13 We will de?ne the mapping class group ¦£1 (M) of any compact C¡Þ manifold M as follows: ¦£1 (M) = Dif f+ (M) /Dif f0 (M) , where Dif f+ (M) is the group of di?eomorphisms of M preserving the orientation of M and Dif f0 (M) is the group of di?eomorphisms isotopic to identity. De?nition 14 Let L ¡Ê H 2 (M, Z) be the imaginary part of a K¡§ ahler metric. We will denote by ¦£2 := {¦Õ ¡Ê ¦£1 (M )|¦Õ(L) = L}. It is a well know fact that the moduli space of polarized algebraic manifolds ML (M) = T (M)/¦£2 . In [7] the following fact was established: Theorem 15 There exists a subgroup of ?nite index ¦£L of ¦£2 such that ¦£L acts freely on T (M) and ¦£\T (M) = ML (M) is a non-singular quasi-projective variety. Over ML (M) there exists a family of polarized CY manifolds ¦Ð : M ¡ú ML (M). Remark 16 Theorem 15 implies that we constructed a family of non-singular CY manifolds ¦Ð : X ¡úML (M) (13)

over a quasi-projective non-singular variety ML (M). Moreover it is easy to see that X ?CPN ¡Á ML (M). So X is also quasi-projective. From now on we will work only with this family. Remark 17 Theorem 15 implies that ML (M) is a quasi-projective non-singular variety. Using Hironaka¡¯s resolution theorem, we can ?nd a compacti?cation ML (M) of ML (M) such that ML (M) ? ML (M) = D is a divisor with normal crossings. We will call D the discriminant divisor.

2.5

A?ne Flat Coordinates around Points at In?nity

Theorem 18 Let U¡Þ = DN ? ML (M) be some open polydisk containing the point ¦Ó¡Þ ¡Ê D. Suppose that U¡Þ ? (U¡Þ ¡É D) = (D? )k ¡Á DN ?k . According to the results proved in [7] there exists a complete family of polarized CY manifolds ¦Ð : X ¡ú U¡Þ ? (U¡Þ ¡É D) = (D? ) ¡Á DN ?k .

k

(14)

Let ¦ØX /U¡Þ be the relative dualizing line bundle. Then there exists a coordinate k k system (¦Ó 1 , ..., ¦Ó k , t1 , ..., tN ?k ) on the universal cover (U) ¡Á DN ?k of (D? ) ¡Á DN ?k and a global section ¦Ø¦Ó ¡Ê ¦£ (U)k , ¦Ð? ¦ØX /(U)k such that:

k k

¦Ø¦Ó = ¦Ø0 +

i=1

¦Øi,0 (n ? 1, 1)¦Ó i +

i¡Üj =1

¦Øij,0 (n ? 2, 2)¦Ó i ¦Ó j + O(3)+

9

n k j =1

N ?k

¦Øj,0 (n ? 1, 1)tj +

i¡Üj =1

¦Øij,0 (n ? 2, 2)ti tj + O(3).

(15)

Proof: The proof of Theorem 18 is based on the results obtained in [6]. Lemma 19 Suppose that ¦Ó¡Þ = 0 ? U¡Þ and D is an open disk in U¡Þ containing 0. Suppose that the monodromy operator T of the restriction of the family (14) on D ? D ¡É D is of in?nite order. Then there exists a non zero section ¦Ø¦Ó ¡Ê ¦£ U¡Þ ? (U¡Þ ¡É D) , ¦Ð? ¦ØX /U¡Þ ?(U¡Þ ¡ÉD) , such that ¦Ø¦Ó = 1 ,

¦Ã0

(16)

where ¦Ã0 a a primitive invariant vanishing cycle with respect to the monodromy operator T. Proof: Let us consider the family (14) . Since we assumed that D ¡É U¡Þ is any open disk and that U¡Þ is a polydisk, then we can construct a non zero family of holomorphic forms ?t over D ¡É U¡Þ according to [6]. So we can analytically extend this family to a family of holomorphic forms ?¦Ó over U¡Þ . Thus we get: ?¦Ó ¡Ê ¦£ ¦Ð ?1 (U¡Þ ? D ¡É U¡Þ ) , ¦ØX /U¡Þ such that at each ¦Ó ¡Ê U¡Þ ? D ¡É U¡Þ , ?¦Ó = 0. According to Theorem 37 proved in [6] we have ?t = 0 and lim

¦Ã0 t¡ú0 ¦Ã0

?t = 0

(17)

for t ¡Ê D. (17) implies that the function ¦Õ(t) =

?¦Ó is di?erent from zero on

¦Ã0

?¦Ó U¡Þ ? D ¡É U¡Þ . Then we can de?ne ¦Ø¦Ó = ¦Õ (¦Ó ) . Clearly the family of holomorphic n-forms ¦Ø¦Ó satis?es (16) . Lemma 19 is proved.

Lemma 20 Suppose that the monodromy operator T of the restriction of the family (14) on D ? D ¡É D = ¦Ó¡Þ is of ?nite order m. Then there exists a n?cycle ¦Ã0 and a non zero section ¦Ø¦Ó ¡Ê ¦£ U¡Þ , ¦Ð? ¦ØX /U¡Þ log D , such that on U¡Þ we have (18) lim ¦Ø¦Ó = 1.

¦Ó ¡ú0 ¦Ã0

Proof: Let ¦Õm : D ¡ú D be the map t ¡ú tm . Let us pullback the restriction of the family (14) by ¦Õm . Then the monodromy operator T of the new family will be the identity. Then we can choose a n?cycle ¦Ã0 such that

t¡ú0 ¦Ã0

lim

?t = 0 .

10

(U¡Þ ¡É D) such that ¦Õ(¦Ó ) := satis?es

The family of holomorphic forms ?t can be prolong to a family ?¦Ó over U¡Þ ? ?¦Ó is a non zero function on U¡Þ . Then ¦Ø¦Ó :=

?¦Ó ¦Õ(¦Ó ) ¦Ã0

¦Ó ¡ú0 ¦Ã0

lim

¦Ø¦Ó = 1

Lemma 20 is proved. We de?ne the ?at a?ne coordinates ¦Ó 1 , ..., ¦Ó k , t1 , ..., tN ?k in (U) ¡Á DN ?k as follows: Let ¦Ø¦Ó be the family of holomorphic n? forms de?ned on U¡Þ by Lemmas 19 and 20. Local Torelli Theorem implies that we can choose a basis of cycles (¦Ã0 , ¦Ã1 , ...., ¦ÃN , ¦ÃN +1 , ..., ¦Ã2N +1 , ...¦Ãbn ) of Hn (M,Z) satisfying ¦Ãi , ¦Ãj

M k

= 0,

¦Ãi , ¦Ã2N +1?j

M

= ¦Äij

for i = 0, ..., k ; j = 1, ..., N ? k such that if ¦Ó i :=

¦Ãi

¦Ø¦Ó , i = 1, ..., k and tj :=

¦Ãj + k

¦Ø¦Ó , j = 1, ..., N ? k

k

(19)

then (¦Ó 1 , ..., ¦Ó k , t1 , ..., tN ?k ) will be a local coordinate system in (U) ¡Á (D)

k

N ?k

.

Lemma 21 Let 0 ¡Ê (U) be any ?xed point. Then the Taylor expansion of the family of holomorphic n forms ¦Ø¦Ó constructed in Lemmas 19 and 20 satis?es (15) . Proof: We know from [14] that we can identify the tangent space at 1,0 0 ¡Ê U¡Þ ? (U¡Þ ¡É D) with H 1 (M0 , TM ). The contraction with ¦Ø0 de?nes an 0 isomorphism 1,0 ?1,0 H 1 M0 , TM ? H 1 M0 , ?n . M0 0 Thus the tangent vectors ¦Õi = ? ?1,0 ¡Ê T0,U¡Þ = H 1 M0 , ?n M0 ?¦Ó i

can be identi?ed with classes of cohomologies ¦Øi,0 (n ? 1, 1) := ¦Ø0 ¦Õi of type (n ? 1, 1). Gri?ths¡¯ transversality implies that for t = 0 we have ? ¦Ø¦Ó ?¦Ó i |¦Ó =0 = a0 ¦Ø0 + ¦Ø0 ¦Õi = a0 ¦Ø0 + ¦Øi,0 (n ? 1, 1) 11 (20)

and

?2 ¦Ø¦Ó ?¦Ó i ?¦Ó j

|¦Ó =0 = (21)

ai,j (0)¦Ø0 + bi,j (0) (¦Øi,j,0 (n ? 1, 1)) + cij (0)¦Øi,j,0 (n ? 2, 2).

Proposition 22 We have a0 = ai,j (0) = bi,j (0) = 0 and cij (0) = const = 0 in the expression (21) . Proof: The de?nition of the coordinates (¦Ó 1 , ..., ¦Ó k ) and (20) and (18) imply that a0 = 0 . (22) From (21) and (22) we can conclude that for 1 ¡Ü i ¡Ü k and t = 0 we have

k

¦Ø¦Ó

k

(U)

k

= ¦Ø0 +

i=1

¦Ó i (¦Ø0 ¦Õi ) +

1 (bij (0)¦Øi,j,0 (n ? 1, 1) + cij (0) (¦Ø0 ¦Õi ¦Õj )) ¦Ó i ¦Ó j + ..., 2 i,j =1 where bij (0) and cij (0) are constants. So (18) implies ? ¦Ø¦Ó = ?¦Ó i

¦Ã0 ¦Ã0

(23)

?2 ¦Ø¦Ó = 0 . ?¦Ó i ?¦Ó j

Since

? ¦Ø¦Ó ?¦Ó i

|¦Ó =0 = ¦Ø0 ¦Õi := ¦Øi,0 (n ? 1, 1),

k

and ¦Ø¦Ó we deduce that

(U)

k

= ¦Ø0 +

i=1

ai ¦Ó i (¦Øi,0 (n ? 1, 1)) + ...

¦Ã0

[¦Øi,0 (n ? 1, 1)] = 0 and

¦Ãj

[¦Øi,0 (n ? 1, 1)] = ¦Äij .

(24)

Thus (24) and (23) imply that ai = 1. The relations (19) and (23) imply that ¦Øi,j,0 (n ? 1, 1) = 0 (25)

¦Ãk

12

for any ¦Ãk such that

¦Ãk

¦Ø¦Ó = ¦Ó k .Indeed (24) implies that for any non zero closed

form ¦Ø (n ? 1, 1) of type (n ? 1, 1) there exists ¦Ãk such that ¦Ø (n ? 1, 1) = 0. (26)

¦Ãk

Thus (25) and (26) imply that bi,j (0) = 0. Proposition 22 is proved. Proposition 22 implies Lemma 21. Lemma 21 implies Theorem 18. Remark 23 Theorem 18 states that the coordinates used in the special geometry and the ?at a?ne coordinates introduced in [14] by Theorems 3 and 7 are the same. This fact is mentioned in #5.1 of [1]. In the same paper the authors referred to [30] (private communication by Tian) for the introduction of the ?at a?ne coordinates.

3

3.1

Metrics on Vector Bundles with Logarithmic Growth

Mumford Theory

In this Section we are going to recall some de?nitions and results from [11]. Let X be a quasi-projective variety. Let X be a projective compacti?cation of X such that X?X= D¡Þ is a divisor with normal crossings. The existence of such compacti?cation follows from the Hironaka¡¯s results. We will look at polydisk DN ? X, where D is the unit disk, N = dim X such that DN ¡É X = (D? )k ¡Á DN ?k , where D? = D ? 0 and q is the coordinate in D. On D? we have the Poincare metric |dq |2 ds2 = 2 2. |q | (log |q |) On the unit disk D we have the simple metric |dt| . The product metric on (D? )k ¡ÁDN ?k we will call ¦Ø (P ) . A complex-valued C¡Þ p-form ¦Ç on X is said to have Poincare growth on X?X if there is a set of if for a covering {U¦Á } by polydisks of X?X such that in each U¦Á the following estimate holds: ¦Ç q 1 , ..., q k , tk+1 , ..., tN where ¦ØU¦Á (q i , q i )

(p) 2

¡Ü C¦Á ¦ØU¦Á (q 1 , q 1 )

2

(p)

2

... ¦ØU¦Á (q k , q k )

(p)

2

(27)

=

2 |q i |

(log |q i |)

|q i |2

2

13

This property is independent of the covering {U¦Á } of X but depends on the compacti?cation X. If ¦Ç1 and ¦Ç2 both have Poincare growth on X?X then so does ¦Ç1 ¡Ä ¦Ç2 . A complex valued C¡Þ p-form ¦Ç on X will be called ¡±good¡± on X if both ¦Ç and d¦Ç have Poincare growth. An important property of Poincare growth is the following: Theorem 24 Suppose that the ¦Ç is a p-form with a Poincare growth on X?X= D¡Þ . Then for every C¡Þ (r ? p) form ¦× on X we have:

X

|¦Ç ¡Ä ¦× | < ¡Þ.

Hence, ¦Ç de?nes a current [¦Ç ] on X. Proof:For the proof see [11]. De?nition 25 Let E be a vector bundle on X with a Hermitian metric h. We will call h a good metric on X if the following holds. 1. If for all x¡Ê X ? X, there exist sections e1 , ..., em ¡Ê E DN ?(DN ¡ÉD¡Þ ) of E which form a basis of E DN ?(DN ¡ÉD¡Þ ) . 2. In a neighborhood DN of x¡Ê X?X in which DN ¡É X = (D? )k ¡Á DN ?k and X?X= D¡Þ is given by t1 ¡Á ... ¡Á tN = 0 the metric hij =h(ei , ej ) has the following properties: a.

k 2m k 2m

hij ¡Ü C

log q

i=1

i

, (det (h))

?1

¡ÜC

log q

i=1

i

(28)

for some C > 0 and m ¡Ý 0. b. The 1-forms (dh) h?1 are good forms onX¡ÉDN . It is easy to prove that there exists a unique extension E of E on X, i.e. E is de?ned locally as holomorphic sections of E which have a ?nite norm in h. Theorem 26 Let (E , h) be a vector bundle with a good metric on X, then the Chern classes ck (E ,h) are good forms on X and the currents [ck (E , h)] represent the cohomology classes ck (E ,h) ¡Ê H 2k X, Z . Proof: For the proof see [11].

14

3.2

Example of a Good Metric

Theorem 27 Let ¦Ð : U1 ¡Á ... ¡Á Uk ¡ú (D? )k be the uniformization map, where U is the unit disk. Suppose that ¦Ê¡Þ ¡Ê (? (U)) = S 1

k k k

,

Let h be a metric on the line bundle L ¡ú (D? ) . Let {¦Óm } ¡Ê U1 ¡Á ... ¡Á Uk be any sequence such that

m¡ú¡Þ

lim ¦Óm = ¦Ê¡Þ ¡Ê U1 ¡Á ... ¡Á Uk ? U1 ¡Á ... ¡Á Uk lim ¦Ð (¦Óm ) = 0 ¡Ê (D) = (D? ) .

k k

and

m¡ú¡Þ

Suppose that ¦Ð ? (h) = hUk is de?ned on U1 ¡Á ... ¡Á Uk as follows:

k

hUk :=

i=1

1 ? |¦Ó i |2 + ¦Õ(¦Ó, ¦Ó ),

k

(29)

where ¦Õ(¦Ó, ¦Ó ) is a bounded C ¡Þ function on (U) and

m¡ú¡Þ

lim ¦Õ(¦Óm , ¦Óm ) = lim hUk (¦Óm , ¦Óm ) = 0.

m¡ú¡Þ k

Then h is a good metric in the sense of Mumford on the line bundle L ¡ú (D? ) . Proof: We need to show that h satis?es the conditions (28) and that ? log hUk is a good form. The conditions (28) followed immediately from the expression (27) for the metric de?ned by hUk . We need to show that h satis?es (27) , i.e. ? log h is a good form. Lemma 28 The (1, 0) form ? log h is a good form. Proof: The de?nition of a good form implies that ? log h is a good form k k on (D? ) if and only i? ? log h(U)k satis?es on the universal cover (U) of (D? )k the following inequalities on each unit disk Ui ? (U) : 0¡Ü and 0¡Ü ? ?¦Ó i

? ?¦Ó i hUi ? ? ?¦Ó i hUi ?¦Ó i hUi k

h Ui

h Ui

¡Üc

1 (1 ? |¦Ó i |2 ) 1 ? |¦Ó i |2 )2

2

(30)

k

h Ui

¡Üc 15

i=1 (1

,

(31)

where c > 0. This statement follows from the fact that the pullback of the metric k k with a constant curvature on (D? ) is the Poincare metric on (U) , i.e.

k

¦Ð

? i=1 |q i |2

dq i

2 2

k

(log |q i |)

=

d¦Ó i

2 2

i=1 (1

? |¦Ó i |2 )

and ? log ¦Ð ? (h) = ? log h(U)k . Proposition 29 The form ? log h(U)k satis?es (30) and (31) . Proof: (30) and (31) will follow if we prove that the restriction of ? log h(U)k on each Ui satis?es (30) and (31) . Direct computations show that the expression (29) of h(U)k implies that we have : hi ¦Ó i , ¦Ó i := h(U)k |Ui = 1 ? |¦Ó |2 + ¦Õi ¦Ó i , ¦Ó i > 0, where

¦Ó i ¡ú¦Êi ¡Þ

(32)

lim ¦Õi (¦Ó i , ¦Ó i ) = lim hi (¦Ó i , ¦Ó i ) = 0

¦Ó i ¡ú¦Êi ¡Þ

and ¦Õi (¦Ó i , ¦Ó i ) is a bounded C ¡Þ function on Ui . (32) implies: 0 ¡Ü 1 ? |¦Ó i |2 ¡Ü Ci hi ¦Ó i , ¦Ó i , (33)

where Ci > 0. Thus we get from (33) that if ¦Ó0 is any complex number such that |¦Ó0 | = 1 then the limit

¦Ó i ¡ú¦Ó0 h

lim

1 ? |¦Ó i |2

i

¦Ó i, ¦Ó i

exists and 0 ¡Ü lim i Direct computations show that ? log hhUi = ?¦Ó i

? ?¦Ó i

¦Ó i ¡ú¦Ó0 h

1 ? |¦Ó i |2

i

¦Ó i, ¦Ó i

= c.

(34)

1 ? |¦Ó i |2 + ¦Õi ¦Ó i , ¦Ó i 1 ? |¦Ó i |2 + ¦Õi ¦Ó i , ¦Ó i

? ?¦Ó i ¦Õi

=

?¦Ó i +

¦Ó i, ¦Ó i . (35)

1 ? |¦Ó i |2 + ¦Õi ¦Ó i , ¦Ó i

We derive from (34) , (35) and the fact that ¦Õi is bounded C ¡Þ function on Ui that we have: ? ? h 1 i h hU i ?¦Ó i Ui ¡Ü c1 0 ¡Ü ?¦Ó 2 h Ui h Ui i (1 ? |¦Ó i |2 ) 16

and

1 (36) 2, (1 ? |¦Ó i |2 ) where c1 > 0. Thus (36) implies that ? log h de?nes a good form on the line bundle L restricted on (D? )k . Lemma 28 is proved. Lemma 28 implies Theorem 27. 0¡Ü ? ?¦Ó i ?hUi h Ui ¡Ü c1

4

4.1

Applications of Mumford Theory to the Moduli of CY

The L2 Metric is Good

We are going to prove the following result: Theorem 30 The natural L2 metric : h(¦Ó, ¦Ó ) = ¦Ø¦Ó

2

:= (?1)

n(n?1) 2

¡Ì ?1

n M

¦Ø¦Ó ¡Ä ¦Ø¦Ó

(37)

on ¦Ð? ¦ØX (M)/ML (M) ¡ú ML (M ) is a good metric. Outline of the proof of Theorem 30. Let (D) be a polydisk in ML (M) N such that 0 ¡Ê (D) ¡É D = ?, where N = dimC ML (M). To prove Theorem 30 we need to derive an explicit formula for the metric ¦Ø¦Ó , ¦Ø¦Ó := h(¦Ó, ¦Ó ) on the line bundle ¦Ð? ¦ØX /ML (M) restricted on (D) ? (D) ¡É D = (D? ) ¡Á (D) Let (Ui )k ¡Á (D)N ?k be the universal cover of (D)N ? (D)N ¡É D = (D? )k ¡Á (D)N ?k , where Ui are the unit disks. Let ¦Ð : (Ui )k ¡Á (D)N ?k ¡ú (D? ) ¡Á (D)

k N ?k N N k N ?k N

.

(38)

be the covering map. We will prove that formula (9) implies that we have the following expression for the L2 metric ¦Ø¦Ó , ¦Ø¦Ó := h(¦Ó, ¦Ó ) on ¦Ð? ¦ØX /ML (M) restricted on the k N ?k universal covering of (Ui )k ¡Á (D)N ?k of (D? ) ¡Á (D) : ¦Ø¦Ó , ¦Ø¦Ó := h(¦Ó, ¦Ó ) :=

k i=1 N

1 ? |¦Ó i |2 +

j =k+1

1 ? |tj |2 + ¦Õ(¦Ó, ¦Ó ) + ¦·(t, t)

(39)

where ¦Õ(¦Ó, ¦Ó ), ¦·(t, t) are bounded real analytic functions on (Ui )k ¡Á (D)N ?k . Theorem 27 and formula (39) imply Theorem 30. Proof : The proof will follow from the Lemma proved bellow. 17

Lemma 31 Let (U) be the universal cover of (D? )k := (D) ? (D) ¡É ML (M) where U is the unit disk. Then i. There exists a family of CY manifolds ¦Ð ? X(U)k ¡ú (U) . over (U) and a holomorphic section ¦Ø ¡Ê H 0 (U) , ¦Ð? ¦ØX (M)/(U)k ¦Ø¦Ó = ¦Ø |M¦Ó is a non zero holomorphic form on M ¦Ó . ii. ¦Ø¦Ó , ¦Ø¦Ó can be represented on (U)k as follows :

k k k k

k

k

k

(40) such that

[¦Ø¦Ó ], [¦Ø¦Ó ] = h(¦Ó, ¦Ó ) =

i=1

1 ? ¦Ói

2

+ ¦Õ(¦Ó, ¦Ó ),

k

(41)

where ¦Õ(¦Ó, ¦Ó ) is a bounded real analytic functions on (U) such that the limits lim exist where

k ¦Ê¡Þ = ¦Ê1 ¡Þ , ..., ¦Ê¡Þ = (q1 ,...,qk )¡ú(0,...,0) ¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

h(¦Ó, ¦Ó ) and

lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

¦Õ(¦Ó, ¦Ó )

lim

¦Ð ?1 (q 1 ) = ¦Ó 1 , ..., ¦Ð ?1 (q 1 ) = ¦Ó k ,

k

(q 1 , ..., q k ) ¡Ê (D? )k , ¦Ð (¦Ê¡Þ ) = (0, ..., 0) ¡Ê D?

k and ¦Ê1 ¡Þ , ..., ¦Ê¡Þ ¡Ê (? U) . h(¦Ó, ¦Ó ) = 0. Then iii. Suppose that lim k ¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k k

= Dk

lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

¦Õ(¦Ó, ¦Ó ) = 0.

k

iv. The function ¦Õ(¦Ó, ¦Ó ) is bounded on U . Proof of i: Since (D? )k ? ML (M), Remark 16 implies that there exists a family of CY manifolds ¦Ð : X(D? )k ¡ú (D? )k (42) over (D? )k . Because p : (U)k ¡ú (D? )k is the universal cover of (D? )k , then the pull back of the family (42) by p de?nes the family (40) . Let ¦Ø¦Ó be the section of the pullback of the restriction of the relative dualizing line bundle ¦ØX /ML (M) on (D? )k on (U)k constructed in Theorem 18. Proof of ii: The proof of Part ii is based on the following Proposition: Proposition 32 Let us consider (D¦Á1 ,¦Á2 ) ? (D? )k ? (D)k ? ML (M), where D¦Á1 ,¦Á2 := {t ¡Ê D¦Á1 ,¦Á2 ||t| < 1 and ¦Á1 < arg t < ¦Á2 } . 18

k

Suppose the closure of (D¦Á1 ,¦Á2 ) in (D)k contains 0 ¡Ê D? consider the restriction of the family (40) X¦Á1 ,¦Á2 ¡ú (D¦Á1 ,¦Á2 ) on (D¦Á1 ,¦Á2 )k ? ML (M). Let ¦ØX

¦Á1 ,¦Á2

k

k

= Dk . Let us

k

(43)

k be the restriction of dualizing (D¦Á1 ,¦Á2 ) k sheaf of the family of polarized CY manifolds (40) on (D¦Á1 ,¦Á2 ) . Then there exists a global section

.

¦Ç ¡Ê ¦£ (D¦Á0 ,¦Á1 ) , ¦Ð? ¦ØX

k

¦Á1 ,¦Á2

.

(D¦Á1 ,¦Á2 )k

such that the classes of cohomology [¦Çq ] de?ned by the restriction of ¦Ç on all of the ?bres ¦Ð ?1 (q ) :=Mq for q ¡Ê (D¦Á1 ,¦Á2 )k are non zero elements of H 0 (Mq , ?n Mq ). The limit lim [¦Çq ] exists and

q¡ú0 q¡ú0 k

lim [¦Çq ] = [¦Ç0 ] and [¦Ç0 ], [¦Ç0 ] ¡Ý 0.

k

(44)

Proof: (D¦Á1 ,¦Á2 ) is a contractible sector in (D? ) . Thus if we ?x a basis (¦Ã1 , ..., ¦Ãbn ) in H n (M,Z)/T or then we are ?xing the marking of the family (43) k over each point M¦Ó for each point ¦Ó ¡Ê (D¦Á1 ,¦Á2 ) . This means that the basis n (¦Ã1 , ..., ¦Ãbn ) of H (M,Z)/T or is de?ned and ?xed on H n (M¦Ó ,Z)/T or on each ?bre of the family (43). Now we can de?ne the period map of the family by ? ? where ¦Çq is a non zero holomorphic form on ¦Ð ?1 (q ) :=Mq . Local Torelli Theorem implies that the period map p of marked CY manifolds p : (D¦Á1 ,¦Á2 ) ¡ú P(H n (M, C)).

k k

p(q ) := ?...,

¦Ãi

¦Çq , ...? ,

(45)

is an embedding (D¦Á1 ,¦Á2 ) ? P(H n (M, C)). Thus we can conclude from the compactness of P(H n (M, C)) and (45) the existence of a sequence of [¦Çq ] such that lim [¦Çq ] = [¦Ç0 ] (46)

q¡ú0

exists and ¦Ç |M0 = 0. (46) implies (44) . Corollary 33 There exists a global section ¦Ç ¡Ê H 0 (U)k , ¦ØX(U)k /(U)k that lim [¦Ç¦Ó ] exists and

¦Ó ¡ú¦Ê¡Þ ¦Ó ¡ú¦Ê¡Þ

such

lim [¦Ç¦Ó ] = [¦Ç¦Ê¡Þ ] = 0.

(47)

19

Proposition 34 Let {¦Ø¦Ó } be the family of holomorphic n?forms constructed in Theorem 18 on the family restricted on (U)k . Then the limit lim [¦Ø¦Ó ] k ¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k exists, = [¦Ø¡Þ ] and [¦Ø¡Þ ] , [¦Ø¡Þ ] ¡Ý 0. (48) lim k ¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k Proof: According to Corollary 33 there exists a global section ¦Ç ¡Ê H 0 (U)k , ¦ØX(U)k /(U)k such that lim [¦Ç¦Ó ] satis?es (47) . The relation between the cohomologies of

¦Ó ¡ú¦Ê¡Þ

holomorphic forms ¦Ç¦Ó := ¦Çq and ¦Ø¦Ó are given by the formula [¦Ç¦Ó ] = ?(¦Ó )[¦Ø¦Ó ], where ?(¦Ó ) is a holomorphic function on the product (U)k . According to Theorem 7 we have (49) 0 ¡Ü [¦Ø¦Ó ], [¦Ø¦Ó ] ¡Ü [¦Ø¦Ó0 ], [¦Ø¦Ó0 ] . Thus (45) , (49) and [¦Ç¦Ó ] = ?(¦Ó )[¦Ø¦Ó ] imply formula (48). So the limit lim exists and lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

[¦Ø¦Ó ], [¦Ø¦Ó ]

[¦Ø¦Ó ], [¦Ø¦Ó ] =

lim h(¦Ó, ¦Ó )

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

= h(¦Ê¡Þ ) ¡Ý 0.

Proposition 34 is proved. Corollary 35 Let [¦Ø¡Þ ] be de?ned by (48) . Then [¦Ø¡Þ ] , [¦Ø¡Þ ] = 0 if and only if the monodromy of the restriction of the family (13) is in?nite. Notice that the functions [¦Ç¦Ó ], [¦Ç¦Ó ] we normalize ¦Ø0 and ¦Õi such that ¦Ø¦Ó0 and ¦Ø0 ¦Õi , ¦Ø0 ¦Õj = ¦Äij we get from (9) the following expression h(¦Ó, ¦Ó ) =

k 2

and [¦Ø¦Ó ], [¦Ø¦Ó ] are real analytic. If

= ¦Ø0 , ¦Ø0 = 1

1?

i=1

|¦Ó i |2 +

i¡Üj

¦Ø0 (¦Õi ¡Ä ¦Õk ) , ¦Ø0 (¦Õj ¡Ä ¦Õl ) ¦Ó i ¦Ó j ¦Ó k ¦Ó l + O(¦Ó 5 ) =

k

1?

i=1

|¦Ó i |2 + ¦µ(¦Ó, ¦Ó ) 20

(50)

holds. Also (50) implies that the restriction of [¦Ø¦Ó ], [¦Ø¦Ó ] = h(¦Ó, ¦Ó ) on the k k universal cover (U) of (D? ) will be given by (41), i.e.

k

[¦Ø¦Ó ], [¦Ø¦Ó ] = h(¦Ó, ¦Ó ) = 1 ?

¦Ói

i=1

2

+ ¦µ(¦Ó, ¦Ó ).

Proof of (41) : We can rewrite the above expression as follows:

k

[¦Ø¦Ó ], [¦Ø¦Ó ] = h(¦Ó, ¦Ó ) = 1 ?

k i=1

¦Ói

i=1 k

2

+ ¦µ(¦Ó, ¦Ó ) =

1? ¦Ó

i 2

? k + 1 + ¦µ(¦Ó, ¦Ó ) =

i=1

1 ? ¦Ói

2

+ ¦Õ(¦Ó, ¦Ó ).

(51)

where ¦Õ(¦Ó, ¦Ó ) = ¦µ(¦Ó, ¦Ó ) ? k + 1. Proposition 34 and (48) imply that lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

h(¦Ó, ¦Ó ),

lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k k

k

¦Õ(¦Ó, ¦Ó )

exist and ¦Õ(¦Ó, ¦Ó ) is a bounded real analytic function on (U) . Part ii of Lemma 31 is proved. Proof of part iii: Suppose that U is the unit disk and lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

¦Ø¦Ó , ¦Ø¦Ó =

k k

lim

¦Ó ¡ú¦Ê¡Þ ¡Ê(U) ?(U)k

k

h(¦Ó, ¦Ó ) = 0.

Notice that since ¦Ê¡Þ ¡Ê U we have 1 ? ¦Êi ¡Þ

2

= 0 and thus

k i=1

? (U) , where U is the unit disk then for each i

1 ? ¦Êi ¡Þ

2

= 0.

(52)

Thus (51) and (52) imply (41) . Part iii is proved. Part iii implies Part iv. Lemma 31 is proved. Lemma 36 Suppose that the L2 metric on the relative dualizing sheaf de?ned by the function h(¦Ó, ¦Ó ) = ¦Ø¦Ó , ¦Ø¦Ó on DN ? DN ¡É D = (D? )k ¡Á DN ?k ? ML (M). is bounded on DN , h|(D)N ?(D)N ¡ÉD > 0 and h|(D)N ¡ÉD ¡Ý 0. Then the L2 metric is good. Proof: The proof of Lemma 36 is obvious. The expression (41) for the L2 metric and Theorem 27 implies that if h|(D)N ¡ÉD ¡Ý 0 then the L2 metric is a good metric . Theorem 30 is proved. 21

4.2

The Weil-Petersson Volumes are Rational Numbers

Theorem 37 The Weil-Petersson volume of the moduli space of polarized CY manifolds is ?nite and it is a rational number. Proof: Theorem 30 implies that the metric on the relative dualizing sheaf ¦ØX /ML (M) de?ned by (37) is a good metric. This implies that the Chern form of any good metric de?nes a class of cohomology in H 2 ML (M), Z ¡É H 1,1 ML (M), Z . See Theorem 26. We know from [14] that the Chern form of the metric h is equal to minus the imaginary part of the Weil-Petersson metric. So the imaginary part of the Weil-Petersson metric is a good form in the sense of Mumford. This implies that ¡ÄdimC ML (M) c1 (h) ¡Ê Z

ML (M)

since ML (M) is a smooth manifold. Since ML (M) is a ?nite cover of the moduli space ML (M) then the Weil-Petersson volume of ML (M) will be a rational number. Theorem 37 is proved. In the paper [10] the authors proved that the Weil-Petersson volumes of the moduli space of CY manifolds are ?nite. Corollary 38 The Weil-Petersson metric is a good metric on the moduli space ML (M) and the Chern forms ck [W. ? P.] of the Weil-Petersson metric are well de?ned elements of H 2k ML (M), Z .

References

[1] M.Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, ¡±Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitude¡±, Comm. Math. Phys. 165 (1994), 311-428. [2] M. Douglas and Z. Lu, ¡±Finiteness of Volume of Moduli Spaces¡±, hepth/0509224. [3] H. Fang and Z. Lu, ¡±Generalized Hodge Metric and BCOV Torsion on Calabi-Yau Moduli¡±, DG/0310007 to appear in Crelle Journal. [4] J. Horne and G. Moore, ¡±Chaotic Coupling Constants¡±, hep-th/9403058. [5] K. Kodaira and Morrow, ¡±Complex Manifolds¡±. [6] B. Lian, A. Todorov and Shing-Tung Yau, ¡±Maximal Unipotent Monodromy for Complete Intersection CY Manifolds¡±, Amer. Jour. of Math. 127(2005) 1-50. 22

[7] K. Liu, A. Todorov, Shing-Tung Yau and Kang Zuo, ¡±Shafarevich Conjecture for CY Manifolds I¡± (Moduli of CY Manifolds), Quarterly Journal of Pure and Applied Mathematics¡±, vol.1 No1 (2005), p. 28-67. [8] Z. Lu, ¡±On the Curvature Tensor of the Hodge Metric of Moduli of Polarized Calabi-Yau Threefolds¡±, J. Geom. Analysis, 11(2001), No 4, 633-645. [9] Z. Lu and X. Sun, ¡±Weil-Petersson Geometry on Moduli space of Polarized Calabi-Yau Manifolds.¡± Jour. Inst. Math. Jussieu (2004) 3(2), 185-229. [10] Z. Lu and X. Sun, ¡±On Weil-Petersson Volume and the First Chern Class of the Moduli Space of CY Manifolds.¡± to appear in Comm. Math. Phys. [11] D. Mumford, ¡±Hirzebruch¡¯s Proportionality Principle in the Non-Compact Case¡±, Inv. Math. 42(1977), 239-272. [12] Y.-T. Siu, ¡±Curvature of the Weil-Petersson Metric on the moduli Space of Compact K¡§ ahler-Einstein Manifolds of Negative First Chern Class.¡± Volume in Honor of W. Stoll. [13] G. Tian, ¡±Smoothness of the Universal Deformation Space of Calabi-Yau Manifolds and its Petersson-Weil Metric¡±, Math. Aspects of String Theory, ed. S.-T. Yau, World Scienti?c (1998), 629-346. [14] A. Todorov, ¡±The Weil-Petersson Geometry of Moduli Spaces of SU(n ¡Ý 3) (Calabi-Yau Manifolds) I¡±, Comm. Math. Phys. 126 (1989), 325-346. [15] A. Todorov, ¡±Witten¡¯s Geometric Quantization of Moduli of CY manifolds¡±, preprint 1999. [16] C. Vafa, ¡±The String Landscape and Swampland¡±, hep-th/0509212. [17] E. Viehweg, ¡±Quasi-Projective Moduli for Polarized Manifolds¡±, Ergebnisse der Mathematik und iher Grenzgebiete 3. Folge, Band 30, SpringerVerlag, 1991. [18] S. T. Yau, ¡±On the Ricci Curvature of Compact K¡§ ahler Manifolds and Complex Monge-Amper Equation I¡±, Comm. Pure and App. Math. 31, 339-411(1979).

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innovative papers [15, 16], Mirzakhani obtained an interesting recursion formula *of* *the* *Weil-Petersson* *volumes* *of* *the* *moduli* *spaces* *of* bordered Riemann ...

On *the* local geometry *of*... 8Ò³ Ãâ·Ñ *The* Weil...Ãâ·Ñ *Weil-Petersson* *Volumes* o... 23Ò³ Ãâ·ÑÈçÒª...bundles on inductive limits *of* *moduli* *spaces* ...

In this paper, we give a simple proof *of* Mirzakhani's recursion formula *of* *Weil-Petersson* *volumes* *of* *moduli* *spaces* *of* curves using *the* Witten-Kontsevich...

Estimates *of* *Weil-Petersson* *volumes* via effective divisors_×¨Òµ×ÊÁÏ¡£We study *the* asymptotics *of* *the* *Weil-Petersson* *volumes* *of* *the* *moduli* *spaces* *of* compact ...

On *the* Geometry *of* *Moduli* *Space* *of* Polarized Calabi-Yau *manifolds* In this ...*volume* and integrability *of* *the* curvature invariants *of* *the* *Weil-Petersson* ...

b are called *the* higher *Weil-Petersson* *volumes* [5]. These are important invariants *of* *moduli* *spaces* *of* curves. In 1990, Witten [6] made *the* ...

polarized *manifolds* induce a hermitian metric on *the* relative canonical bundle...and generalized *Weil-Petersson* metrics and hyperbolicity *of* *moduli* *spaces*. 1...

Recently in [1] it has been shown that *the* instanton *moduli* *spaces* *of* ...ed by *the* instanton contributions and by *the* *Weil-Petersson* *volume* forms ...

We proved *the* *Weil-Petersson* metric and *the* new...cations *of* *the* *moduli* *spaces* *of* holomorphic maps ...Conjecture for Odd Dimensional *CY* *Manifolds*. Joint...

We consider *the* *moduli* *space* *of* *the* extremal K\"ahler metrics on compact *manifolds*. We show that under *the* conditions *of* two-sided total *volume* bounds...

- Weil-Petersson volumes of moduli spaces of curves and the genus expansion in two dimensiona
- Moduli spaces of hyperkaehler manifolds and mirror symmetry
- The Weil-Petersson Geometry On the Thick Part of the Moduli Space of Riemann Surfaces
- Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds
- The homotopy type of the space of symplectic balls in rational ruled 4-manifolds
- Geometry of the Weil-Petersson completion of Teichmuller space
- Recursion between Mumford volumes of moduli spaces
- Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces
- Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models
- On the Geometry of Moduli Space of Polarized Calabi-Yau manifolds