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arXiv:math/0505583v2 [math.DG] 2 Jun 2005

ON THE CURVATURE TENSOR OF THE HODGE

METRIC OF MODULI SPACE OF POLARIZED

CALABI-YAU THREEFOLDS

ZHIQIN LU

Contents

1.

2.

3.

4.

References

Introduction

The Covariant Derivatives of the Yukawa Coupling

The Estimates

A Remark on the Theorem of C-L Wang

1

3

9

11

13

1. Introduction

This paper is the continuation of the paper [6] of our study of the Moduli

space of polarized Calabi-Yau threefold.

A polarized Calabi-Yau manifold is a pair (X,ω) of a compact algebraic

manifold X with zero first Chern class and a K¨ ahler form ω ∈ H2(X,Z).

The form ω is called a polarization. Let U be the universal deformation

space of (X,ω). U is smooth by a theorem of Tian [12]. By [15], we may

assume that each X′∈ U is a K¨ ahler-Einstein manifold. i.e. the associated

K¨ ahler metric (g′

αβ) is Ricci flat. The tangent space TX′U of U at X′can

be identified with H1(X′,TX′)ωwhere

H1(X′,TX′)ω= {φ ∈ H1(X′,TX′)|φ?ω = 0}

The Weil-Petersson metric GPW on U is defined by

?

where φ = φγ

β

α

is the K¨ ahler-Einstein metric on X associated to the polarization ω.

Let n = dimU. As showed in [6], we defined the Hodge metric ωHby

GWP(φ,ψ) =

X′g′αβg′

γδφγ

βψδ

αdVg′

∂

∂zγdzβ, ψ = ψδ

∂

∂zδdzα∈ H1(X′,TX′)ω, and g′= g′

αβdzαdzβ

ωH= (n + 3)ωWP+ Ric(ωWP)

where ωWP is the K¨ ahler form of the Weil-Petersson metric.

The main result of [6] is the following

Theorem 1.1. Let ωH= (n + 3)ωWP+ Ric(ωWP). Then

Date: Dec. 14, 1997.

1

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(1) ωHis a K¨ ahler metric on U;

(2) The holomorphic bisectional curvature of ωH is nonpositive. Fur-

thermore, Let α = ((√n + 1)2+ 1)−1> 0. Then the Ricci curvature

Ric(ωH) ≤ −αωH and the holomorphic sectional curvature is also

less than or equal to −α.

(3) If Ric(ωH) is bounded, then the Riemannian sectional curvature of

ωHis also bounded.

The main result of this paper builds on the above theorem:

Theorem 1.2. Let X be a Calabi-Yau threefold.

orthonormal harmonic basis of H1(X,TX)ω. Then there is a constant C,

depending only on n, such that the L∞norm of the sectional curvature |R|

satisfies

n

?

Remark 1.1. The crucial part of this theorem is that the curvature has an

upper bound which only depends on the L4norm of the harmonic basis,

rather than depends on the derivative of the harmonic basis. Upper bound

of the sectional curvature of the Hodge metric is very important in the

compactification of the moduli space (cf. [5]).

Let ϕ1,··· ,ϕn be the

|R| ≤ C

i=1

||ϕi||8

L4

In order to prove the theorem, we need to estimate the covariant derivative

of the Yukawa coupling with respect to the Weil-Petersson metric. As a by-

product, we proved the following theorem (for definitions, see §2):

Theorem 1.3. Let F = (Fijk) be the Yukawa coupling. Let

Fijk,l= ∂lFijk− Γm

Then Fijk,l= Fijl,k.

ilFmjk− Γm

jlFimk− Γm

klFijm+ 2KlFijk

Remark 1.2. The moduli space of a Calabi-Yau threefold is a projective

special K¨ ahler manifold in the sense of D. Freed [2]. In [4], the Yakuwa

coupling of special K¨ ahler manifolds is discussed.

The motivation behind this paper and the paper [6] is that we want to

give a differential geometric proof of the theorem of Viehweg [13] in the case

of the moduli space of Calabi-Yau threefolds. Viehweg’s theorem states

that moduli spaces of polarized algebraic varieties are quasi-projective. The

boundedness of the curvature of the Hodge metric is very important because

of the work of Mok [7], Mok-Zhong [8] and Yeung [16]. By their theorems,

if a complete K¨ ahler manifold of finite volume has negative Ricci curvature

and bounded sectional curvature, then it must be quasi-projective. In the

case of the moduli space of Calabi-Yau threefolds, the Ricci curvature is

negative (Theorem 1.1), and the condition on the boundedness of the sec-

tional curvatures can be weakened, thus it is very important to get various

upper bound estimates of the sectional curvatures.

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In the last section, we give an extra restriction on the limit of Hodge

structures for a one dimensional degeneration of a family of Calabi-Yau

threefolds.

Acknowledgment. The author thanks Professor Tian for his constant

encouragement and discussion during the preparation of this paper. He also

thanks C. L. Wong for a lot of useful conversations.

2. The Covariant Derivatives of the Yukawa Coupling

Suppose π : X → U is the total space over the (local) universal defor-

mation space U of a Calabi-Yau threefold X. Thus for any point X′∈ U,

π−1(X′) is a Calabi-Yau threefold. The Hodge bundle F3= π∗ωX/Uis the

holomorphic line bundle over U where ωX/Uis the relative canonical bundle

of X.

There is a natural Hermitian metric on F3defined by the Ricci flat metric

on each fiber of π. Such a metric can be written out explicitly as follows:

since for any X′∈ U, π−1(X′) is differmorphic to π−1(0) = X, there is a

natural identification of H3(X′,C) → H3(X,C). Suppose ϕ,ψ ∈ H3(X,C).

Define the cup product

Q(ϕ,ψ) = −

?

X

ϕ ∧ ψ

Let Ω be a local holomorphic section of F3. Thus for each X′∈ U, Ω

at X′is a holomorphic (3,0) form on H3,0(X′) ⊂ H3(X′,C) and under the

identification H3(X′,C) → H3(X,C), Ω(X′) ∈ H3(X,C).

The Hermitian metric on F3is defined by setting ||Ω||2=√−1Q(Ω,Ω).

The technical heart of this paper is to compute the covariant deriva-

tive of the Yukawa coupling with respect to the Weil-Petersson metric and

the Hermitian metric on the bundle F3. Recall that by definition(see[1],

for example), the Yukawa coupling is the (local) section F of the bun-

dle Sym3((R1

H1(X′,TX′) and Ω ∈ H3,0(X′),

?

Here TX/Uis the relative tangent sheaf of X → U.

The basic property of the Yukawa coupling is that it is a holomorphic

section. In fact, Let t1,··· ,tnbe the local holomorphic coordinate system

of U. Let Ω be a local nonzero section of the holomorphic bundle F3, We

have

Fijk= F(ρ(∂

∂tk)) = Q(Ω,∂i∂j∂kΩ),

where ρ : TXU → H1(X,TX) is the Kodaira-Spencer map.

Let Γk

ijbe the Christoffel symbols of the K¨ ahler metric gijand let Kl=

−∂llog||Ω||2be the connection of the Hermitian bundle F3with respect to

the local section Ω. We make the following definition:

∗(TX/U))∗) ⊗ (F3)⊗2over U such that for any ϕ1,ϕ2,ϕ3 ∈

F(ϕ1,ϕ2,ϕ3) =

X′(ϕ1∧ ϕ2∧ ϕ3?Ω) ∧ Ω

∂ti),ρ(∂

∂tj),ρ(∂

1 ≤ i,j,k ≤ n

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Definition 2.1. For 1 ≤ i,j,k ≤ n, the covariant derivative of Fijk is

defined as

Fijk,l= ∂lFijk− Γs

liFsjk− Γs

ljFisk− Γs

lkFijs+ 2FijkKl

In this section, we are going to compute Fijk,lat a point X ∈ U in terms

of the information of the fixed Calabi-Yau threefold X.

We use the method developed by Siu[11], Nannicini[9] and Schumacher[10].

By K¨ ahler geometry, there is a holomorphic coordinate (t1,··· ,tn) of U such

that at X, Γk

is carefully chosen, then Kl= 0, 1 ≤ l ≤ n at X.

Consider the Kodaira-Spencer map ρ : TX′U → H1(X′,TX′). Let ϕj=

ρ(∂

∂tj),1 ≤ j ≤ n. Suppose ϕj’s are harmonic TX′-valued (0,1) forms. These

ϕj’s can be realized by the canonical lift in the sense of Siu [11] (See also

Nannicini [9] and Schumacher [10]): suppose (z1,z2,z3) is the holomorphic

coordinate on X. Then for each

∂tj, there is a vector vj, called the canonical

lift of

∂tj, on X which locally can be represented as vj=

that ϕj= ∂vα

j

It should be noted that vj is a vector field on X but neither is

vα

j

is different from it is as the vector field on U. It is also easy to check

that the real part of the vector field vj defines differmorphisms between

the fibers. Using these differmorphisms, tensor fields of the nearby fibers

can be identified as tensor fields on X. By Nannicini [9] or Siu [11], the

Lie derivative Lvl· is defined as the usual

differmorphisms.

Now let’s analyze the conditions Kl= 0 and Γi

at X. We have

ij= 0, 1 ≤ i,j,k ≤ n. Furthermore, if the local section Ω of F3

∂

∂

∂

∂tj+ vα

j

∂

∂zα such

∂

∂zα is a harmonic TX′-valued (0,1) form.

∂

∂tj nor

∂

∂zα alone. In fact, the component

∂

∂tjin the expression vj=

∂

∂tj+ vα

j

∂

∂zα

∂

∂tl after pulling back via the

jk= 0 for 1 ≤ i,j,k,l ≤ n

Proposition 2.1. We use the notations as above. In particular, suppose

(∂

∂t1,··· ,

Then we have

(1) (LvlΩ)3,0= 0;

(2) (Lvkϕi) is a ∂∗-boundary.

∂

∂tn) and Ω are chosen such that Γi

jk= 0 and Kl= 0 at X ∈ U.

Proof: The key point is to identify the derivatives with respect to co-

homological classes and the derivatives with respect to forms. Suppose for

fixed l, vl= τ1+√−1τ2where τ1and τ2are real vector fields. Let σ1(s) and

σ2(s) be the flows defined by τ1and τ2, respectively. Consider the 3-forms

σi(s)∗Ω, i = 1,2. Suppose

σi(s)∗Ω = pi(s) + dqi(s)

be the Hodge decomposition of σi(s)∗Ω in H3(X,C). Then we have

ds|s=0(p1(s) +√−1p2(s))

0 = ∂lΩ =

d

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by the definition of ∂lΩ. This is equivalent to

d

ds|s=0(σ1(s)∗Ω +√−1σ2(s)∗Ω) − dd

Or in other word

ds|s=0(q1(s) +√−1q2(s)) = 0

LvlΩ − dσ = 0

for σ =

Using this, we have

d

ds|s=0(q1+√−1q2).

?

X

Ω ∧ LvlΩ = 0

On the other hand, Kl= 0 implies

0 = ∂lQ(Ω,Ω) =

?

X

LvlΩ ∧ Ω +

?

X

Ω ∧ LvlΩ

So the first part of the proposition follows from the following:

Claim. ∂LvlΩ = 0.

Proof of the Claim: This follows from a straightforward computation.

Let Ω be represented as

Ω = adz1∧ dz2∧ dz3

where the functions a,z1,z2,z3are holomorphic on each fiber and have

parameter t. Suppose ρ(∂

form. We have

∂(∂

∂t(dz1∧ dz2∧ dz3))

= ∂d∂z1

∂t

= −∂∂∂z1

∂t

= (−∂1ϕ1

By the harmonicity of ϕ, we have

∂αϕα

∂t) = ϕ = ϕα

β∂αdzβis a harmonic TX-valued (0,1)

∧ dz2∧ dz3+ ∂(dz1∧ d∂z2

∧ dz2∧ dz3+ dz1∧ ∂∂∂z2

β− ∂2ϕ2

∂t

∧ dz3) + ∂(dz1∧ dz2∧ d∂z3

∧ dz3− dz1∧ dz2∧ ∂∂∂z3

β)dz1∧ dz2∧ dz3∧ dzβ

∂t)

∂t ∂t

β− ∂3ϕ3

(2.1)

(2.2)

β+ Γα

αγϕγ

β= 0

where the notation Γα

connection Γi

From the theory of deformation of complex structures, we know that ∂−tϕ

defines the ∂-operator on the nearby fibers. Thus we have

∂∂a

∂t= ϕα

Using Equation (2.1), (2.2) and (2.3), we have

βγis the connection of X which is different from the

jkon the universal deformation space U.

(2.3)

β∂αadzβ

∂∂

∂tΩ = ∂∂a

= (ϕα

∂tdz1∧ dz2∧ dz3+ a∂∂

β∂αa − aΓα

∂t(dz1∧ dz2∧ dz3)

αγϕγ

β)dzβ∧ dz1∧ dz2∧ dz3