Page 1
arXiv:math/0505583v2 [math.DG] 2 Jun 2005
ON THE CURVATURE TENSOR OF THE HODGE
METRIC OF MODULI SPACE OF POLARIZED
CALABIYAU 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 CL 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 CalabiYau threefold.
A polarized CalabiYau 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¨ ahlerEinstein 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 WeilPetersson metric GPW on U is defined by
?
where φ = φγ
β
α
is the K¨ ahlerEinstein 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 WeilPetersson 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 CalabiYau 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
ϕi8
L4
In order to prove the theorem, we need to estimate the covariant derivative
of the Yukawa coupling with respect to the WeilPetersson 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 CalabiYau 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 CalabiYau threefolds. Viehweg’s theorem states
that moduli spaces of polarized algebraic varieties are quasiprojective. The
boundedness of the curvature of the Hodge metric is very important because
of the work of Mok [7], MokZhong [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 quasiprojective. In the
case of the moduli space of CalabiYau 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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3
In the last section, we give an extra restriction on the limit of Hodge
structures for a one dimensional degeneration of a family of CalabiYau
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 CalabiYau threefold X. Thus for any point X′∈ U,
π−1(X′) is a CalabiYau 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 Ω2=√−1Q(Ω,Ω).
The technical heart of this paper is to compute the covariant deriva
tive of the Yukawa coupling with respect to the WeilPetersson 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 KodairaSpencer map.
Let Γk
ijbe the Christoffel symbols of the K¨ ahler metric gijand let Kl=
−∂llogΩ2be 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 CalabiYau 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 KodairaSpencer 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 3forms
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
dss=0(p1(s) +√−1p2(s))
0 = ∂lΩ =
d
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by the definition of ∂lΩ. This is equivalent to
d
dss=0(σ1(s)∗Ω +√−1σ2(s)∗Ω) − dd
Or in other word
dss=0(q1(s) +√−1q2(s)) = 0
LvlΩ − dσ = 0
for σ =
Using this, we have
d
dss=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 TXvalued (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
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So ∂∂
∂tΩ = 0 follows from the fact
aΓα
αγ= a∂γloga2= a∂γloga = ∂γa
and the claim is proved.
The second part of the proposition is implied in [9]. We prove it for the
sake of completeness. We assume at X, (z1,z2,z3) are normal coordinates.
By definition,
Lvkϕi= (∂k(ϕi)α
β− (ϕi)γ
β∂γvα
k)∂αdzβ
Thus
(2.4)
∂∗Lvkϕi= (∂β∂k(ϕi)α
β− (ϕi)γ
β∂β∂γvα
k)∂α
We are going to prove ∂∗Lvkϕi= 0. By the harmonicity of ϕi, we have
gβ1β(ϕi)α
β,β1= 0
or
gβ1β(∂β1(ϕi)α
β+ Γα
β1β2(ϕi)β2
β) = 0
Taking derivative with respect to ∂kgives
(2.5)
gβ1β∂∂kβ1(ϕi)α
β+ ∂kgβ1β∂β1(ϕi)α
β+ ∂k(gβ1βΓα
β1β2)(ϕi)β2
β= 0
Since Lvkω = 0 (See Nannicini [9], for example), we have
0 = Lvkgββ1dzβ∧ dzβ1= (∂kgββ1+ ∂βvβ1
So we have
∂kgββ1= −∂βvβ1
We also have
k)dzβ∧ dzβ1
(2.6)
k
(2.7)
∂k(gβ1βΓα
β1β2) = ∂kΓα
ββ2= ∂kΓβαβ2= ∂k∂β2gβα= −∂β2∂βvα
k
In addition, we have
(2.8)
∂k∂β1(ϕi)α
β= ∂β1∂k(ϕi)α
β− ∂β1vγ
k∂γ(ϕi)α
β
Using Equation (2.6), (2.7) and (2.8), from Equation (2.4) and (2.5), we
get ∂∗Lvkϕi= 0.
By [9], we see that
0 = Γijk=
?
X
< Lvkϕi,ϕj>
So the harmonic part of Lvkϕiis zero. Thus Lvkϕiis a ∂∗boundary by
the Hodge decomposition.
?
The condition that (∂
ϕ1,··· ,ϕn∈ H1(X,TX) is a set of orthnormal basis of harmonic TXvalued
forms. Let Ω be the local nonzero section of F3. We make the following
definition:
∂t1,··· ,
∂
∂tn) is an orthonormal basis implies that
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Definition 2.2.
ajk= (ϕj∧ ϕk)#= ϕj∧ ϕk?Ω
is an (1,2) form for 1 ≤ j,k ≤ n. Here
ϕj∧ ϕk= (ϕj)α
If Ω = adz1∧ dz2∧ dz3. Then ajkcan be represented as
(2.9)
ajk= a(ϕj)α
β(ϕk)γ
δ
∂
∂zα∧
∂
∂zγ⊗ dzβ∧ dzδ
β(ϕk)γ
δsgn(ξ,α,γ)dzξ∧ dzβ∧ dzδ
Lemma 2.1. For 1 ≤ j,k ≤ n,
∂∗ajk= 0
Proof: Since X is a K¨ ahler manifold, we have ∂∗=√−1[Λ,∂]. First
we have ∂ajk= 0. Next, suppose (z1,z2,z3) is a normal coordinate system.
Then
Λajk= Λ(a(ϕj)α
β(ϕk)γ
δsgn(α,γ,β)dzδ− a(ϕj)α
δsgn(α,γ,ξ)dzξ∧ dzβ∧ dzδ)
= a(ϕj)α
β(ϕk)γ
β(ϕk)γ
δsgn(α,γ,δ)dzβ
However, the fact that ϕj?ω = 0 and ϕk?ω = 0 implies (ϕj)α
and (ϕk)γ
β= (ϕj)β
α,
δ= (ϕk)δ
γ. So Λajk= 0 and the lemma is proved.
?
Definition 2.3. The Hodge *operator on X is defined as
∗ : Ap,q→ An−p,n−q,
(ϕ,ψ)dV = ϕ ∧ ∗ψ
for ϕ,ψ ∈ Ap,q(X).
Lemma 2.2. For 1 ≤ j,k ≤ n
∗ajk= ajk
Proof: By Equation (2.9),
ajk= a(ϕj)α
β(ϕk)γ
δsgn(α,γ,ξ)dzξ∧ dzβ∧ dzδ
We have
∗ ajk=
?
β(ϕk)γ
m<n
a(ϕj)α
β(ϕk)γ
δsgn(α,γ,ξ)sgn(ξ,m,n)sgn(β,δ,η)dzm∧ dzn∧ dzη
= a(ϕj)α
δdzα∧ dzγsgn(β,δ,η)dzη= ajk
Here we again use the fact ϕj?ω = 0 and ϕk?ω = 0.
?
Lemma 2.3. For 1 ≤ j,k ≤ n,
∂(Lvjϕk?Ω) = ∂ajk
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Proof: In Nannicini [9], it is proved that
∂Lvjϕk= D∗
1(ϕj∧ ϕk)
where
D∗
1(ϕj∧ ϕk) = ∂α((ϕj)α
β(ϕk)γ
δ)∂γdzβ∧ dzδ
The lemma follows from Equation (2.9).
Now we are going to prove the main theorem of this section.
Theorem 2.1. Let G be the Green’s operator on differential forms of X.
Suppose ϕ1,··· ,ϕnare the orthnormal basis of H1(X,TX). Then
?
where aij’s are defined in Definition 2.2.
Fijk,l=
X
(G∂ali,∂ajk)dV +
?
X
(G∂alj,∂aik)dV +
?
X
(G∂alk,∂aij)dV
It should be noted that the notation of the inner product is defined as
(a,a) = a2. So (a,a) is not the norm of a.
Proof: Since we choose the local coordinate (t1,··· ,tn) and the local
section Ω such that Γi
jk= Kl= 0, we have Fijk,l= ∂lFijk. By Proposi
tion 2.1,
∂lFijk= ∂l
?
X
(ϕi∧ ϕj∧ ϕk)?Ω) ∧ Ω =
?
X
(Lvlϕi∧ ϕj∧ ϕk?Ω) ∧ Ω
+
?
X
(ϕi∧ Lvlϕj∧ ϕk?Ω) ∧ Ω +
?
X
(ϕi∧ ϕj∧ Lvlϕk?Ω) ∧ Ω
Thus we need only to prove that
?
X
((Lvlϕi) ∧ ϕj∧ ϕk?Ω) ∧ Ω =
?
X
(G∂ali,∂ajk)dV
Let Lvlϕi= bli. Recall that Ω = adz1∧ dz2∧ dz3. Then
(2.10)(bli?Ω) = (−1)αa(bli)α
By Equation (2.9) we see that
α1dz1∧ ···ˆ
dzα··· ∧ dz3∧ dzα1
(bli?Ω ∧ ajk) = (−1)αa2(bli)α
sgn(β,γ,ξ)dzξ∧ dzβ1∧ dzγ1
= a2(bli)α
α1(ϕj)β
β1(ϕk)γ
γ1dz1∧ ···ˆ
dzα··· ∧ dz3∧ dzα1
α1(ϕj)β
β1(ϕk)γ
γ1sgn(α,β,γ)sgn(α1,β1,γ1)
dz1∧ dz2∧ dz3∧ dz1∧ dz2∧ dz3
= −(bli∧ ϕj∧ ϕk?Ω) ∧ Ω
By Lemma 2.2 and the above equation, we have
(bli∧ ϕj∧ ϕk?Ω) ∧ Ω = −(bli?Ω ∧ ajk)
= (bli?Ω ∧ ∗ajk) = (bli?Ω,ajk)dV
(2.11)
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By Lemma 2.3 and Proposition 2.1, we have
2(Lvlϕi?Ω) = ∂∗∂(Lvlϕi?Ω) + ∂∂∗(Lvlϕi?Ω)
= ∂∗∂ali+ ∂((∂∗Lvlϕi)?Ω) = ∂∗∂ali
Since Lvlϕiis a ∂∗boundary (Proposition 2.1), we know
Lvlϕi?Ω = G∂∗∂ali
(2.12)
where G is the Green operator of the Laplacian 2. Thus by Equation (2.11)
and (2.12),
?
=
X
((Lvlϕi) ∧ ϕj∧ ϕk?Ω) ∧ Ω =
?
?
X
(bli?Ω,ajk)dV
X
(G∂∗∂ali,ajk) =
?
X
(G∂ali,∂ajk)dV
?
Corollary 2.1 (Theorem 1.3). For 1 ≤ i,j,k,l ≤ n,
Fijk,l= Fijl,k
?
3. The Estimates
In this section, we give an upper bound of the curvature tensor of the
Hodge metric. We use the same notations as in the previous section.
Suppose (z1,··· ,zn) is the normal holomorphic coordinate system at p ∈
U with respect to the WeilPetersson metric ωWP=√−1gijdzi∧ dzj.
We further assume that Q(Ω,Ω) = 1. In [6] we have proved that
Theorem 3.1. If (z1,··· ,zn) is the normal coordinate system of ωWP, then
the curvature tensor Rijklof ωH=√−1hijdzi∧ dzjat p is
Rijkl= Aijkl+ Bijkl
where
Aijkl= 2δijδkl+ 2δilδkj− 4
?
(
?
s
FiksFjls+ 2
?
?
mnpq
FqkmFplmFinpFjnq
Bijkl=
?
rs
Firs,kFjrs,l−
?
mnrs
Firs,kFmrs)(
rs
Fjrs,lFnrs)hnm
Here Fijkis the Yukawa coupling and Fijk,lis its covariant derivative with
respect to the WeilPetersson metric and the connection on F3.
It is proved in [6] that in order to bound the curvature tensor, we need
only to bound the scalar curvature. By definition, the scalar curvature ρ is
ρ = −hijhklRijkl= −hijhklAijkl− hijhklBijkl
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Lemma 3.1. Suppose that the dimension of the universal deformation space
U is n. Then
hijhklAijkl ≤ 3n6
Proof: Under the local coordinate (z1,··· ,zn), we proved in [6] that
hij= 2δij+
?
Suppose further that
?
Then
hij= (2 + λi)δij
In particular, for fixed i,m,n
mn
FimnFjmn
mn
FimnFjmn= λiδij
Fimn ≤
?
λi
So
?
=
ijkl
hijhkl?
?
?
mnpq
1
2 + λi
FqkmFplmFinpFjnq
ikmnpq
·
1
2 + λkFqkmFpkmFinpFinq
≤
ik
1
2 + λi
·
1
2 + λkλiλkn4
In [6], we have proved that hijhklAijkl≥ 0. Thus
hijhklAijkl ≤
ik
?
4
(2 + λi)(2 + λk)+ 2
?
ik
1
2 + λi
·
1
2 + λkλiλkn4≤ 3n6
?
Now we consider hijhklBijkl. It is easy to see that
(3.1)0 ≤ hijhklBijkl≤
?
ijklrs
hijhklFirs,kFjrs,l≤
?
ijkl
Fijk,l2
Lemma 3.2. Using the notations in Theorem 2.1 , we have
Fijk,l ≤ 3(ϕi4
L4 + ϕj4
L4 + ϕk4
L4 + ϕl4
L4)
Proof: Since the Green operator is a positive operator, we have

?
X
(G∂ali,∂ajk) ≤
??
X
(G∂ali,∂ali)
??
X
(G∂ajk,∂ajk)
However, for fixed l,i, by Lemma 2.1, we have
G∂∗∂ali= G2ali= ali− H(ali)
where H(ali) is the harmonic part in the Hodge decomposition of ali. Thus
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?
X
(G∂ali,∂ali) ≤ 
?
X
(G∂∗∂ali,ali) ≤
?
X
ali2≤
?
X
ϕl4+ ϕi4
So by Theorem 2.1,
Fijk,l ≤ 3(ϕi4
L4 + ϕj4
L4 + ϕk4
L4 + ϕl4
L4)
?
Theorem 3.2 (Theorem 1.2). The scalar curvature ρ of the Hodge metric
satisfies
0 < −ρ ≤ 3n6+ 144n3?
Proof: ρ < 0 follows from [6]. The upper bound is from Lemma 3.1 and
Lemma 3.2.
i
ϕi8
L4
?
4. A Remark on the Theorem of CL Wang
In his paper [14], CL Wong gave a necessary and sufficient condition
for the WeilPetersson metric to be incomplete for a family of CalabiYau
manifolds over a punctured disk. The main theorem of him is (for the precise
definitions and notations, see Wong [14]):
Theorem 4.1 (CL Wong [14]). Let ∆∗be the parameter space of a family
of CalabiYau manifolds. Let Fn
theory and N is the associated nilpotent operator. Then the necessary and
sufficient condition for the WeilPetersson metric to be incomplete is that
NFn
∞= 0.
∞be the limit of Fnin the sense of Hodge
In this section, we are going to prove, even if the WeilPetersson metric
is complete, we still have some restrictions on Fn
The classical WeilPetersson metric is defined by giving a natural Hermit
ian metric on H1(X,TX) induced by the Ricci flat K¨ ahler metric for each
CalabiYau manifold. However, by the theorem of Tian [12], We can look
at the WeilPetersson metric in a different way.
Recall that the Hodge bundle Fnover the classifying space D is the
tautological bundle of the filtration
∞for n = 3.
0 ⊂ Fn⊂ Fn−1⊂ ··· ⊂ F1⊂ H
The natural Hermitian metric on Fnis the polarization Q. Suppose ω is the
curvature form of the Hermitian metric Q, then ω is an closed (1,1) form of
D. Suppose M is a horizontal slice of D (see Griffiths [3], for example), then
ω restricts to a semipositive form on M. However, if M is the universal
deformation space of a CalabiYau manifold, then by Tian’s theorem [12],
ωMmust be positive definite and is the WeilPetersson metric.
Thus there are some restrictions for a horizontal slice on which the ω is
positive definite. The following theorem gives one of the restrictions on the
limiting Hodge structure.
Page 12
12
Theorem 4.2. We use the notations in the above theorem and in [14]. If
n = 3, then
Q(F3
∞,N3F3
∞− 3N2F3
∞− 2NF3
∞) = 0
Proof: Let
Ω = e
1
2π√−1logzNA(z)
where A(z) is a vector valued holomorphic function of z ∈ ∆∗, the punctured
unit disk. Let
Fzzz= (Ω,∂z∂z∂zΩ)
It is easy to check that
lim
z→0z3Fzzz= Q(F3
∞,N3F3
∞− 3N2F3
∞− 2NF3
∞)
So we need only to prove that
lim
z→0z3Fzzz= 0
Let p ∈ ∆∗. Then since p represents a CalabiYau threefold, we have a
map f from a neighborhood of p in ∆∗to the universal deformation space U.
Suppose in local coordinates, the map f is z ?→ (z1,··· ,zn). Let Zi=∂zi
Then from [6], we see that the Hodge metric on ∆∗can be written as
∂z.
h = hijZiZj= (2gij+ gmngpqFimpFjnq)ZiZj
where gijand hijare the WeilPetersson metric and the Hodge metric, re
spectively. Since gij≤ hij, we have
h ≥ hmnhpqFimpFjnqZiZj
By the Cauchy inequality, we see that
(hmnhpqFimpFjnqZiZj)h2≥ FijkZiZjZk2
So we have
h3≥ FijkZiZjZk2= Fzzz2
In [6], it is proved that the curvature of h is negative away from zero. So
the Schwartz lemma gives,
h ≤
C
r2(log1
r)2
Then
r6Fzzz2≤ C
r6
r6(log1
r)6→ 0
The theorem is proved.
?
Page 13
13
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Invent. Math 106 (1991), 13–25.
(Zhiqin Lu) Department of Mathematics, Columbia University, New York,
NY 10027
Email address: lu@math.columbia.edu