The Journal of Geometric Analysis Volume 1 I, Number 1, 2001

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau

Threefolds B y Zhiqin Lu

1. Introduction

A polarized Calabi-Yau manifold is a pair (X, 09) of a compact algebraic manifold X with zero first Chern class and a K/ihler form 09 ~ HZ(X, Z). The form o9 is called a polarization. Let 34 be the universal deformation space of (X, o9). 34 is smooth by a theorem of Tian [5]. By [8], we may assume that each X / E 34 is a Kahler-Einstein manifold! i.e., the associated K~.hler metric ( g ~ ) is Ricci flat. The tangent space Tx,34 of 34 at X ' can be identified with H I ( x ', Tx,)o~

where H I (X t, Tx,)~ o = {cb 6 H 1 (X t, Tx,)[q~Ao9 = 0 } .

The Weil-Petersson metric Gpw on A/[ is defined by

f t t~ r y--8 Owe(r r = Jx, g

where q~ • 0 ~ 0 = q$~ 0-~-d2~, ~P = ~p~g~z ~ d ~ are in H 1 (X', Tx,)~o, g' = g 1 dzC~ d-it~ is the K~ le r -

Einstein metric on X associated with the polarization w.

In this article, we consider the universal deformation space 34 of a simply connected Calabi-Yau threefold. Let wwe be the K~laler form of the Weil-Petersson metric and set n = dim H I ( x , Tx) for some X 6 34. We proved

Theorem 1.1. LetwH = (n + 3)o9wp + Ric(ogwp). Then

1. w14 is a K~ihler metric on AA ;

2. The holomorphic bisectional curvature of WH is nonpositive. Furthermore, Let ot = ( ( x / ~ + 1) 2 + 1) -1 > 0. Then theRiccicurvatureRic(wH) < -OtWl4 and the holomor- phic sectional curvature is also less than or equal to -or.

3. I f Ric(o9 H ) is bounded, then the Riemannian sectional curvature of o9 H is also bounded.

Key Words and Phrases. harmonic map, rigidity theorem, symmetric space.

�9 2001 The Journal of Geometric Analysis ISSN 1050-6926

104 Zhiqin Lu

Because of the following theorem, we call ~On the Hodge metric of the universal deformation space. For the definitions, see Section 2 and Section 3.

Theorem 1.2. Let U be an open neighborhood of J~, and let U --+ D be the period map to the classifying space D. Then up to a constant, o914 is the pull back of the invariant Hermitian metric of the classifying space D.

R e m a r k 1.3. In fact, we have proved more. The theorems are also true on the normal horizontal slices. A normal horizontal slice is a horizontal slice such that the Weil-Petersson metric can be defined. See Section 3 for details.

The proof of the first theorem is a straightforward computation using the Strominger's for- mula [4]. Using this method, we can also find the optimal upper bound of the Ricci curvature and the holomorphic sectional curvature. The combination of the first and the second theorem is somewhat unexpected. Let's explain this a little bit more in detail. By a theorem of Griffiths, we know that the holomorphic sectional curvature on the horizontal directions of the classifying space is negative away from zero. Using the same method, we know that the holomorphic bisectional curvature is nonpositive on certain directions. I f D is a homogeneous Kdhler manifold, then by the Gauss theorem, we should be able to prove that the holomorphic sectional curvature and the holomorphic bisectional curvature of the horizontal slice are smaller than the corresponding cur- vatures on the classifying space. However, D is not a homogeneous Kahler manifold in general. Nevertheless, the theorems tell us that we still have the negativity of the curvatures.

In order to prove the second theorem, we make use of the fact that D is the dual homogeneous manifold of a Kahler C-space. Write D = G~ V where G is a noncompact semi-simple Lie group without compact factors and V is its compact subgroup. Let K be the maximal compact subgroup containing V. We write out explicitly the projection G~ V ~ G / K via local coordinate. Then the metric (n + 3)~owp + Ric(ww p) and the restriction of the invariant Hermitian metric of D on U can be identified.

In the last section, we gave an asymptotic estimate of the Weil-Petersson metric to the degeneration of Calabi-Yau threefolds. Such an estimate was obtained by Tian [6] in the case that the degenerated Calabi-Yau threefold has only ordinary double singular points. C-L. Wang also got such a result using a completely different method.

2. The classifying space and the horizontal slices

The concepts of the classifying space and the horizontal slice were introduced by Griffiths [2]. We recall his definitions and notations in this section.

Suppose X is a simply connected algebraic Calabi-Yau three-fold. The Hodge decomposition of the cohomology group H = H3(X, C) is

H3(X, C) : H 3'0 ~ H 2'1 G H 1'2 �9 H 0'3

where

H p'q = H q (X, ~P)

and ~P is the sheaf of the holomorphic p-forms. The quadratic form Q on X is defined by

Q ( ~ , r l ) = - f x ~ A ~ .

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds 105

By the Serre duality and the fact that the canonical bundle is trivial, dim H 2' 1 = dim H 1,2 = dim H i ( x , Tx) = n, and dim H 3'~ = dim H ~ = 1. Thus, H3(X, C) = C 2n+2 is a (2n+2)-

dimensional complex vector space.

It is easy to check that Q is skew-symmetric. Furthermore, we have the following two Hodge-Riemann relations:

1. Q ( H p'q, H p''q') --- 0 unless pt = 3 - p and qt = 3 - q;

2. (~-----1)P-q Q ( ~ , ~-) > 0 for any nonzero element ~ 6 H p'q .

We define the Weil operator C " H --~ H by

For any collection of { g p'q }'s, set

F 3 = H3, 0

F 2 = H3, 0 @ H 2,1

F l = H 3'~ G H 2'1 (9 H 1'2 �9

Then F 1 , F 2, F 3 defines a filtration of H

0 C F 3 C F 2 C F 1 C H .

Under this terminology, the Hodge-Riemann relations can be re-written as

3. Q ( F 3 , F 1) =O, Q ( F 2 , F 2) = 0 ;

4. Q ( C ~ , ~ ) > 0 i f ~ r

Now we suppose that {h p'q } is a collection of integers such that p + q = 3 and Y~ h p'q =

2n + 2 .

Def in i t ion 2.1. With the notations as above, the classifying space D of the Calabi-Yau three- fold is the set of all collection of subspaces { H p'q } of H such that

H = (9 H p'q H p'q = H q , P , dim H p'q = h p'q p+q=3

and on which Q satisfies the two Hodge-Riemann relations 1,2.

Set f P = h n'O q- . . �9 d- h p'n-p. Then D is also the set of all filtrations

0 C F 3 C F 2 C F 1 C H, F p (9 F 4-p = H

with dim F p = f P on which Q satisfies the bilinear relations 3,4.

D is a homogeneous complex manifold. The horizontal distribution Th (D) is defined as

TtT(D) = [ X ~ T ( D ) I X F 3 C F 2, X F 2 C F 1 ]

where T ( D ) is the holomorphic tangent bundle which can be identified as a subbundle of the (locally trivial) bundle H o m ( H 3 (X, C), H 3 (X, C)). So X naturally acts on FP.

106

Definition 2.2. a horizontal slice.

Zhiqin Lu

A complex integral submanifold of the horizontal distribution Th (D) is called

Suppose U C .M is a neighborhood of .M at the point X. Then there is a natural map p : U --+ D, called the period map, which sends a Calabi-Yau threefold to its "Hodge Structure" To be precise, let X' 6 U. Then there is a natural identification of H3(X t, C) to H3(X, C) = H. So { n p 'q (Xt)}p+q=3 are the subspaces of H satisfying the Hodge-Riemann Relations. We define p (X t) = {HP'q(xt)} E D.

3. The Weil-Petersson metric and the Hodge metric

On the classifying space D, we can define the so-called Hodge holomorphic bundles F 3, F 2, F 1, which are the subbundles of the locally trivial bundle C 2n+2. The fiber of the bundle C 2n+2 at X 6 .AA is H 3 (X, C). The fibers of __F 3, _F 2, _F_.E 1 at X are H 3,0 (X), H 3,0 (X) E) H 2' 1 (X), H3,~ ~ H2,1(X) ~ H1,2(X), respectively. Note that F 3 is in fact a line bundle. Let f2 be a (nonzero) local holomorphic section of F 3. The curvature form of the bundle F 3 is then

cr = - C 00 log a (f2, ~) .

Let U be a horizontal slice, define

CO = f f l U �9

Proposition 3.1. Let w = - ~ g a ~ d z a A d-~ 13 in local coordinate. Then ga-~ > 0 is semi- positive definite.

Proof. Let K = - log Q(f2, ~ ) . Then

Q (0uf2 + KaY, o'9/~ f2 + K S~)

But 0uf2 + Kuf2 ~ H 2,1 . The proposition follows from the second Hodge-Riemann Relation. [ ]

Definit ion 3.2. The horizontal slice is called normal if the form o9 is positive definite at any point. In that case, co = ogwe is called the Weil-Petersson metric on the normal horizontal slice.

R e m a r k 3.3. By the theorem of Tian [5], we know that i f .M is a universal deformation space, then ~r IA4 is the Weil-Petersson metric defined in the introduction. Thus the universal deformation space is a normal horizontal slice.

Definition 3.4. The cubic form F = Fijk is a (local) section of the bundle Sym3(T*.M) | (F3) | defined by

F i j k = a (f2, O i O j O k ~ )

in local coordinates (z 1 , . . . , zn).

Definition 3.5. Let U be a normal horizontal slice. Suppose ojo is the Kahler form of the invariant Hermitian metric on D, then we call o9o I u the Hodge metric on U.

We are going to prove the following theorem.

Theorem 3.6. Suppose o9w P is the Kiihler form of the Weil-Petersson metric. Let

o91 = (n + 3)o9we + Pdc(~we)

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds

then o21 is a constant multiple o f the Hodge metric.

Before proving the theorem, we first prove the following.

Proposition 3.7. There is a basis el, " ' " , e2n+2 Of H under which Q can be represented as

107

1 )

Jr t : D --+ C P 2n+l

is the projection of D to C P 2n+l by sending (F 3, F 2, F 1) to F 3. Let U be a normal horizontal slice. Let f2 be a (nonzero) local section of F.__33. Then Oi ~2 -~- gi ~ is not zero because ogwp is positive. Thus

Jr I : U ~ D ~ C P 2n+l

is an immersion.

Now we consider the result of Bryant and Griffiths [1]. Their results can be briefly written as follows.

We assume that eV E U i.e., the normal horizontal slice passes the original point of D, where the original point is defined as {f3, f2, f l} E D in the Proposition 3.7. Then according to Bryant and Griffiths, there is a holomorphic function u defined on a neighborhood of the original point of C n such if (z 1, . . . , z n) is the local holomorphic coordinate of U at eV, the original point, then

( --~2 ~ 1 i 1 ~ / - ~ l u n ~ ] (3.1) ~ = 1 , z l , " ' " , zn, u - - E ~ Z Ui,----~Ul,''" , /~ i

Clearly, (hi]) > O.

Suppose

The point {f3, f2, f l } E D is called the original point of D. Sometimes we write it as eV i fD = G / V .

By the curvature formula of Strominger [4], the Ricci curvature of the Weil-Petersson metric

Ri] = - ( n + 1)gi] + e2K FipqFjmngpm g qn

where we set w w p = Cz-T~ t, ij - dzi A d-z j in the local coordinates (z 1 , �9 .. , z n) and K is the local

function: K = - log Q(f2, f2).

Letwl = ---~-hi]dz i A d-~ j . Then

hi] = 2gi] + e2K FipqFjmngpm g qn .

And i f we let

f 3 = span { el -- V/-~en+2 }

f 2 = s p a n [ e l - - ~ / ~ e n + z , e z + V - Z - T e n + 3 , ' " ,en+l + x/~e2n+2}

and f l is the hyperplane perpendicular to f 3 with respect to Q, then

{ 0 c f3 c f 2 c f l c H } E D .

108 Zhiqin Lu

with F l = Span{(S2)}, F 2 = span{VS2}, F 1 • F 3 via Q. In particular, u(0) = -4"-L-] -, IVu(0)l = 0, V2u(0) = 4 ~ I , where I is the unit matrix.

In order to prove Theorem 3.6, we need only to prove it at the original point, because any point of the homogeneous space D can be taken as the original point.

U n d e r t h e s e notations, at e V, Di ~ ~- Oi ~2 q- K i ~ = ~2 (ei+l + ~-~en+i+2) . T h u s

VcZ]O (f2, ~ ) = - 2

and Q ( D i a , Dj~--)

gij -- 0 ( ~ 2 , ~ )

Furthermore, the cubic form Fijk at eV is

Thus

1 = " ~ i j .

1 O3u 1 Fij~ -- 2 Ozi OzJOz ~ (0) = -~u i j~ (O) .

- - 1 h i j = 2gi] + e 2K FimnFjpqgmp g nq = Sij + ~Uimn(O)u-~mn(O) .

Now we are going to prove that (hi~) is a constant multiple of the Hodge metric. Consider the projection

yr " D = G / V ~ G / K

where K is the maximal connected compact subgroup of G containing V. We have.

L e m m a 3.8. Let U be a horizontal slice, then 7r is an isometry between the Riemannian submanifold U o f D and the Riemannian submanifold 7r (U) oF G / K.

Proof . Note that U is a horizontal slice of D. The lemma follows from the definition of the invariant Hermitian metric on both manifolds. [ ]

From the above lemma, we know that in order to compute O)DIU, w e need only computed the metric of U as a submanifold of G / K , even the map Jr is not holomorphic (recall that D is not homogeneous K~ihler, so the map will not be holomorphic in general). In order to do this, we write out the projection

rc : G / V -+ G / K

explicitly now.

It is easy to prove from linear algebra that the projection zr send

0 C F 3 C F 2 C F 1 C H

to F 3 ~ H 1,2 .

We have known that G / K = Sp (n + 1, R ) / U (n + 1 ) is the Hermitian symmetric sp ace. G / K can be realized as the set of (n + 1) planes P in the C 2n+2 space such that - ~ / Z T Q ( P , -fi) > O. Thus G / K can be represented as the set of all the symmetric (n + 1) • (n + 1) matrix Z satisfying Im Z > 0 where Im Z > 0 means Im Z is a positive definite Hermitian matrix.

We write the entries of the matrix Z as functions of D.

On the Hodge Metric of the Universal Deformation Space of Calabi- Yau Threefolds 109

Suppose now near the original point, F 3 is spanned by

(1, z t, a, ot t)

where z, c~ 6 C n, a c C. And suppose F 2 is spanned by the row vectors of the matrix

( l zt a ~ ) 0 1 /3 A

for fl c C n, A ~ fl[(n, C). Then by the first Hodge-Riemann relation

Q (F2, F 2) = 0

we know that fi = ct - A z, A t = A .

So locally, we can represented F 2 by the matrix

( 1 zt a ~ ) A t = A (3.2) 0 1 ot - Az A

Let f2 ----- (1, z t, a, ot'), and let (~) be a local section of F 2 with | = (0, 1, c~ - Az, A) . Set

m = Q (f2, ~ = - a + -d - oltz -{- ~tz

= a (S2, ~ ) = - ~ + ff - A (-~ - z)

where m 6 C, ~ c C n. It is easily checked that

So O - - m ~-~ E F 1 and since f2 and | are in H 2'1, ~ - ~ 6 H 1'2.

The projection re can be locally written as

0 1 c~ - Az A > -0 - ~

( 1 z t a ott )

- m 1 - ~ - c t - A z - - f f - A - m

where the right-hand side of the above represents an (n + 1)-plane in H.

The symmetric (n + 1) x (n § 1) matrix Z can be obtained as follows: let

1

# ---- m - (~t _ z t ) ~

Then as a matrix ( ,)' 1 = l - ' m

We have

-- "~ m --Z t B

1 ~ t = B L B - - m - - ~ m

110 Zhiqin Lu

where B = 1 + / z~ (2 t - zt). Let

( 1 - z t B - ~ - z t B ) ( a ott ) ( D1 Dt2 ) n ~ B ot- ,z~-a--~ m X - ~ - f f ' = 03 04

for D1 ~ C, D2, D3 E C n, D4 E l~[(n, C). Then it can be computed

01 : a - z t (or - -Az) -q- /z (zt~) 2

D2 = D3 = (a - ~) Iz~ -1- B(ot - Az) (3.3)

D4 = A + / z ~ t

Then the matrix Z is obtained.

Proposition 3.9. Under the notation as above, the map

Jr : G / V ~ G / K

under the local coordinate described as above is

( 1 zt a ~ ) ( D1 D~ ) = Z 1 u - A z A > D3 D4

where the Di 'S ale defined as in Equation (3.3).

The Hermitian metric on G / K is - ~ L 1 0 0 log det Im Z. In particular, at the original point,

it is Y~4j d Z ij A d Z ij, where we set Z ij = Z ji if i > j . By Equation (3.1) we see that in

order to get the map U --~ G/K, z, a, or, A in Equation (3.3) should be replaced by A z , u -

~zl iul., -'~2 VU, ~2 Uij, respectively. Thus we have

L~t = 0, L0-x = 0 Otk Otk

O(Da)r - - ~ i ~ r k , O(D3)r - - 0 (3.4) Otk - - Ot k

OD4 O(D4)rs = O, Otk - - -Ursk

By a straightforward computation, we know the restriction of the metric on G/K on U at the original point is a constant multiple of

1 hi7 : t~ij -'[- -~Uimn(O)u-'~mn(O) �9

Thus completes the proof.

4. The curvature computation

In this section we give an optimal estimate of the upper bound of the holomorphic sectional curvature, bisectional curvature, and the Ricci curvature of a normal horizontal slice.

Let U be a normal horizontal slice. Suppose (gi]) is the Weil-Petersson metric, (Fijk) is the

cubic form, and K = - log Q([2, f2). The Hodge metric (hi]) is:

hi] : 2gi] q- Z e2K FirsFjPqgrpgsq " rspq

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds 111 ~

As we have proved, (hi]) is a K~ihler metric. So the curvature tensor Ri-]k i of (hi]) is

~ 02hi] hn m Ohi~ Ohnj R i j k 7 - Ozk02l OZ k 021 "

Now we suppose that at point p, the local coordinate for the Weil-Petersson metric is normal i.e., at point p, gi-] ---- ~ i j and dgi- ] ---- 0. Furthermore, assume K(p) = 0. The curvature tensor Ri]ki of (gi-]) then is

02gi] . Ri]ki = OzkO-~l '

we also have

3hi~ ~Z k -- Z girs,kFmrs

r s

(4.1)

where

Firs,k = OkFirs + 2KkFirs

is the covariant derivative of the cubic form with respect to the Weil-Petersson metric. By using the Strominger formula at p,

We get

Ri]ki = ~ijSkl "-1-~il~kj -- FikmFjlm �9

02hi] OzkOg l = 2Ri]ki -- 2 E Rq ~kTFirs Fjrq

sqr

-}- 28kl Z Firs Firs q- Z f ir s'k fjrs,l . rs rs

(4.2)

Combining Equation (4.1) and Equation (4.2), we have the following.

Proposition 4.1. I l K = 0 at the point p,

gi-]ki = 2Ri]ki + 2r~kl Z FirsFjrs -- 2 y ~ Rq-gkiFirsFjr q rs sqr

+ Z f i r s ' k f j r s ' l - - Z ( ~ r s f i r s ' k fmrs ) (~rs f j r s ' l f n r s ) mn

(4.3)

Based on the above proposition, we get the following theorem.

T h e o r e m 4.2. Letc(n) = ( (~ /n+ 1) 2 + 1), then

1 RiC(WH) < c(n)O)H

1 R < - - - -

- c ( n )

where R is the superium of the holomorphic sectional curvature. The constant here is optimal. Furthermore, the bisectional curvature is nonpositive.

112 Zhiqin Lu

P r o o f We consider the point p and the normal coordinate at p with respect to the Weil-

Petersson metric. Fixing i, let

Am = ~ Firs,kak Fmrs rsk

for a vector a = (al, �9 .. , an). Then it is easy to see that

Fipq, kak 2 , .) h n~ ~pq Z k --~mn (r~sk firs 'kakfmrs) (r~sk firskakfnrs

~ k ~mn f n p q 2 ~ a ~ m 2 = ~pq Fipq,ka k -- hnmAm + 2 hamAm

(4.4)

where we use the fact that hi7 = 2~ij + Fimn Fjmn at p.

Define a generic vector a k = '~ik, k = 1, �9 .. , n. Using Equation (4.4), we have

pq mn

Now using Proposition 4.1, we get

Rc~yg ac~aeayaa = RiTd > 2eiiii q- 2 Z IFirs 12 - 2 ~ Rq~i~FirsFir q rs *qr

> 4 - 4 Z IFiirl2 -k- 2 Z Fqip~irq �9 r rp

Let

Then we have

x = y ~ [Fiir] 2 . r

rp q Fq ip~ i rq 2 ~q 2 ~_, Y ~ >_ IFiiql 2 = x 2

~ - . Z 1 2 fqip~i~q > IFqirl 2 > - (hi~ - 2) .

n rp q

So for a, b > 0, a + b = 1, we have

1 - b ~Ri~i- [ ~ 2 - 2x + ax 2 -}- - (hi7 - 2) 2

n

1 b > 2 - - + - (h d - 2) 2 .

f/ n

On the other hand, we have m

hc~aa afl = h d �9

Let a =

On the Hodge Metric of the Universal Deformation Space of Calabi- Yau Threefolds

2+,/a 2+2,/ff and b = 1 - a, we have

113

1 h 2 _ 1 hiTai-s f 2 [{iii7 > (V ~ +1)2 + 1 " c(n)

It is a straightforward computation that the constant here is optimal.

Thus we proved/~(a , ~, a, ~) > [lall 2, since a can be any vector by making a linear trans- formation of the normal coordinate. We have already proved the assertion of the theorem about the holomorphic sectional curvature.

Now we turn to the bisectional curvature. For any (a 1 , �9 �9 �9 , an), using the same inequalities before, we have

2

RiTkia k a 7 > 2 Y~k ak 2 + 2 a i2 -- 4 ~_.r ~ Firkak

Fqkm~irqa k 2 a i 2 ~r ~q k --ar 2 + 2 Z Z > 2 + 2 Fiqk~iqra k > 0 . mr qk

This proves the nonpositivity of the bisectional curvature.

Finally we consider the Ricci curvature. Suppose that ~ is a unit vector. definition of the Ricci curvature and above results, we have

Then by the

-R ic (~ , ~) > /~ (~, ~-, r ~) .

This completes the proof of the theorem. []

5. The boundness of the sectional curvature

In this section, we prove that the boundness of the Ricci curvature implies the boundness of the Riemannian sectional curvature.

Theorem 5.1. Suppose U is a normal horizontal slice. Suppose p E U is a fixed point such that the R3"cci curvature has a lower bound Cp at p. That is

Ric (WH)p ~_ - C p (WH)p .

Then the Riemannian sectional curvature has a bound

/~(X, Y, X, Y) _< (3 + Cp)IlXll2llYII 2

where X, Y ~ Tp U and X 3_ Y.

We begin by restating Proposition 4.1 in Section 4.

Proposition 5.2. Suppose we have the notations as h7 the Proposition 4.1, then we have

Rijkl : Ai]ki "1- Bi]kl

114 Zhiqin Lu

where

Ai~ki = 2~ijSkl -}- 28ilSkj -- 4 y ~ FiksFjl s q- 2 Z FqkmFplmFinpFjnq s mnpq

Bi~k,----~rs(firs,k-~mnaik~fnrshn'~)(fjrs,l-~mnajl -~fnrshn~)

+ 2 y ~ Aik~hnmAjl~lh ml~ , mmln

and Aikm = Z Firs'k Fmrs "

r s

Proof. A straightforward computation.

Lemma 5.3. (1 O) " - Suppose that,, r/E T~ ' U and define I1~11 = = h i ] ~ j, then

gi~ki~i~ko-~ l < (6 + Cp)11~112110112 .

[ ]

Proof Because the holomorphic bisectional curvature of U is nonpositive. We know that the holomorphic sectional curvature is bounded by Cp, i.e.,

eiTki~i~k~--~ l ~ Cpll~lt 4 �9

We have

where we use the fact that

pq ~nij Finp~jnq~i-~f 2

:~pq ~n (~i finp~i) (~j fjnqrlJ) 2

< Z " FjnqrlJ Pq

-- ~ i F i 2 ~j Fjnqrl j 2 -E ,~ Z pn qn

_< 11~11211oll 2

and

hit = 2~ij + Z Firs Fjrs rs

2

i,j pn

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds 115

Thus

We also have

E Z FqkmFplmFinpCnq~i~kojol ijkl mnpq

(g ) =~pq finpfjnq ~i-~ (~klfqkmfplm~rl)

~ i~pq [~nij FinpFjnq~iO'--f[2~pq ~mkl Fqkm~p lm~k-~2

< 11~112110112 .

~ik Fiks~i~ k 2 (~k ~i Fiks~ i 2 ) _ I1~114 sZ Thus by Proposition 5.2, we have

E aijki~i~ko-~ l < 611~11211r1112 . ijkl [

We also have

Z Bij k'~i~k~-~l ~ , / Z niTk[~i~k~J~----~Z niTk'TliokojlTl" ijkl Vijkl ijkl

Combining the above two inequalities we proved the lemma.

Proof of Theorem 5.1. Let

q = x - J - ~ J X

= y - C L - T j y .

Then 1 (Re/~ (~, ~, ~, ~) - / ~ (~, ( , O, O)) J~(X, Y, X, Y) = ~

The bisectional curvature is bounded by Cp

R (~,~-, rl,~) _< Cpll~ll2110112 �9

Thus

g ( x , Y, X, Y) _< al (3 +Cp)11~112110112 = (3 + Cp)IlXll211YII 2. D

116 Zhiqin Lu

6. A n a s y m p t o t i c e s t i m a t e

In this section, we make use of the results in the previous sections to prove an asymptotic estimate of the Weil-Petersson metric of the degeneration of Calabi-Yau threefolds. The motiva- tion for the estimate is from a result of G. Tian [6]. Although the argument can be generalized to study the Weil-Petersson metric of a normal horizontal slice near infinity, we restrict ourselves to the degeneration of Calabi-Yau threefolds.

We say re : 3~ --~ A is a degeneration of Calabi-Yau threefolds, if 3~, A are complex manifolds and re is holomorphic, and A is the unit disk in C. 'r ~ A, t # 0, re-1 (t) is a smooth Calabi-Yau threefold while re-1 (0) is a divisor of normal crossing. We also denote A* to be the punctured unit disk.

T h e o r e m 6.1. Suppose 3~ -~ A is a degeneration o f Calabi- Yau threefolds. Suppose w be the Weil-Petersson metric on A*. Then i f

Then

lim log ~o = 0 . r~0 log 1

[ 1 ",~ 4c(n) 09 __< w/-~C1 [klog r ) dz A d-z

where c ( n ) = ((v/-ff + 1) 2 + 1), C1 is a constant and z is the coordinate o f A .

R e m a r k 6 . 2 . w is a K ~ l e r metric on A*. So there is a function )~(z) > 0 on A* such that

w = ~ ) ~ ( z ) d z A d-~.

The assumption is understood as log )~

lim , = 0 r -+ 0 log r

and the conclusion of the theorem is understood as

( ! ) 4 c ( n ) ~(z) < CI log

Proof . By a theorem of Tian, there are no obstructions towards the deformation of a Calabi- Yau three-fold. Suppose M 6 r e - l (A*) is a fiber. Let n = dim H i ( M , | and re(M) = p. Let .A4 be the universal deformation space at p. Then there is a neighborhood U of A* at p such that U C .M Suppose Zl = z, and suppose U is defined by z2 . . . . . Zn = 0 near the point p. Using the same notations as in the previous sections, by the Strominger's formula, we have

RlllT = 2g~T 1 (~ , -~ )2 Fllse Fllo g ~ " (6.1)

where (f2, ~ ) = ~#Z1Q(f2, ~ ) . []

L e m m a 6.3 . Let )~ = g ff , then

~ - 1 1 1 . - - - - ~ ( a , ~ ) ~ Fll~ Fllog 6F <- ( a , ~ )2 Flac Fl~ogU~ g r

On the Hodge Metric of the Universal Deformation Space of Calabi-Yau Threefolds 117

Proof If gl~ -- 0 for fl # 1, then the inequality is trivially true. Thus we would like to choose a coordinate such that g l? -- 0, fl # 1.

Let A be an (n - 1) x (n - 1) matrix. Let

tOi =- Z A i j z j j--2

for i = 2, �9 �9 - , n. If A is a nonsingular matrix, then ( Z l , W 2 , " �9 �9 , W n ) will be local holomorphic coordinate of A/f at p and A* is again be defined by w2 . . . . . wn = 0. Now we choose an A such that

s = o

for k = 2, ... , n. Suppose ~ is the matrix under the coordinate (zl, w2.." , Wn), (gl~) = 0

forot # 1.

Using the lemma, from Equation (6.1), we have

RIIIT >_ 2Z 2 - Z (hi t - 2glT ) > - Z h l T .

a2 then the Gauss curvature of A* with respect to ~. is Suppose A = aZlaZl'

4 K = - - - A log~..

On the other hand,

So we have

1 A log log - =

r 4r2 (log l ) 2 '

1 1 A l o g Z = - ~ Z K _> ~R1T1T _> - h i t

where we use the Gauss formula 4Rf f f f < - K & 2. The holomorphic sectional curvature of (hiT) is less than 1 - c - ~ " Thus by the Schwartz lemma

c(n)

( r log 1) 2"

Thus,

A log > 0 . (log 1)4c(n) -

The rest of the proof is quite elementary: let

f = log (log 1) 4c(n)

then

lim f r~01og r 1-

= 0 .

118 Zhiqin Lu

I large enough. Now So for any 6, there is a ~ such that r < 8 implies - f + �9 log ;

A ( - - f + 6 l o g ! ) < 0 .

So the minimum point must be obtained at Ir I = �89 Therefore,

1 f - e log - < C1 -I- E log 2 < 2C1 ,

r

for any e small, thus letting e ---, 0, we have f < 2C1, which completes the proof. [ ]

R e m a r k 6.4. In Hayakawa [3], the author claimed a relation between the degeneration of the Calabi-Yau manifolds and the noncompleteness of the Weil-Petersson metric. But her proof was incomplete. C-L. Wang [7] gave a proof of this and studied the Weil-Petersson metric in great detail. In particular, he proved an asymptotic estimate for the degeneration of Calabi-Yau manifold which is slightly sharper then our estimate independently using a different method.

Acknowledgment

This paper is a refinement of a part of my Ph.D. thesis. The author would like to thank his advisor, Professor G. Tian for his advise and constant encouragement during my four-year Ph,D. study. He also thanks Professor S.T. Yau for his constant encouragement and many impor- tant ideas stimulating further study of this problem.

References

[1] Bryant, R. and Griffiths, P. Some Observations on the Infinitesimal Period Relations for Regular Threefolds with Trivial Canonical Bundle. In Arithmetic and Geometry, Artin, M. and Tate, J., Eds., 77-85, Birldaaiiser, Boston, 1983.

[2] Griffiths, P., Ed. Topics in Transcendental Algebraic Geometry. volume 106 of Ann. Math. Studies. Princeton University Press, (1984).

[3] Hayakawa, Y. Degeneration of Calabi-Yau Manifold with Weil-Petersson Metric. Technical Report alg- geom/9507016, Okolahoma State University, July 1995.

[4] Strominger, A. Special Geometry. Comm. Math. Phy., 133, 163-180, (1990).

[5] Tian, G. Smoothness of the Universal Deformation Space of Compact Calabi-Yau Manifolds and its Peterson-Weil Metric. In MathematicalAspects of String Theory, Yau, S.-T, Ed., volume 1,629-646, Wodd Scientific, 1987.

[6] Tian, G. Smoothing 3-folds with Trivial Canonical Bundle and Ordinary Double Points. In Essays in Mirror Symmetry, Yau, S.-T, Ed., 458-479, International Press, 1992.

[7] Wang, C.-L. On the Weil-Petersson Metrics and Degeneration of Calabi-Yau Manifolds. Technical report, Harvard University, December 1996.

[8] Yau, S.-T. On the Ricci Curvature of a Compact K~ihler Manifold and the Complex Monge-Ampere Equation, I. Comm. Pure Appl. Math., 31,339-411, (1978).

Received July 20, 1998

Department of Mathematics, University of California, Irvine, CA 92697 e-mail: zlu@math.uci.edu