Geometry of CY moduli (II) - Home | UCI Mathematicszlu/talks/2010-KIAS/kias-2.pdf · Geometry of CY moduli (II) Zhiqin Lu 陆志勤 KIAS, 2010 Department of Mathematics, UC Irvine,

Geometry of CY moduli (II)

Zhiqin Lu陆志勤KIAS, 2010

Department of Mathematics,UC Irvine, Irvine CA 92697

June 19, 2010

Zhiqin Lu, Dept. Math, UCI Geometry of CY moduli (II) 1/39

Chern classes– A quick review

We assume that E → M is a holomorphic vector bundle over acompact complex manifold M.

Let h be a Hermitian metric on E. Let

Γ = ∂h · h−1.

be the matrix of the connection.Let

R = ∂Γ

be the curvature matrix.

We assume that E → M is a holomorphic vector bundle over acompact complex manifold M.Let h be a Hermitian metric on E. Let

Γ = ∂h · h−1.

be the matrix of the connection.

LetR = ∂Γ

We assume that E → M is a holomorphic vector bundle over acompact complex manifold M.Let h be a Hermitian metric on E. Let

Γ = ∂h · h−1.

be the matrix of the connection.Let

R = ∂Γ

Let f be an invariant polynomial. That is, f is a polynomial onCr2 such that

f(A) = f(T−1AT).

The Chern-Weil form with respect to the polynomial f isdefined by f(R).

Let f be an invariant polynomial. That is, f is a polynomial onCr2 such that

f(A) = f(T−1AT).

The Chern-Weil form with respect to the polynomial f isdefined by f(R).

If in addition, we assume that f has integer coefficients, thenwe have the following results:

...1 f(R) is closed. This is the Chern-Weil form;

...2 [f(R)] ∈ H∗(M,Z), and is independent of the choice ofthe connection. Gauss-Bonnet-Chern Theorem

If in addition, we assume that f has integer coefficients, thenwe have the following results:

...1 f(R) is closed. This is the Chern-Weil form;

...2 [f(R)] ∈ H∗(M,Z), and is independent of the choice ofthe connection. Gauss-Bonnet-Chern Theorem

This talk is based on the joint work with MichaelR. Douglas

Zhiqin Lu and Michael R. DouglasGauss-Bonnet-Chern theorem on moduli spacespreprint, 2008, arXiv:0902.3839.

TheoremLet f be an invariant polynomial with rational coefficients. LetR be the curvature tensor with respect the Weil-Peterssonmetric. Then ∫

is a rational number.

This talk is based on the joint work with MichaelR. Douglas

Zhiqin Lu and Michael R. DouglasGauss-Bonnet-Chern theorem on moduli spacespreprint, 2008, arXiv:0902.3839.

TheoremLet f be an invariant polynomial with rational coefficients. LetR be the curvature tensor with respect the Weil-Peterssonmetric. Then ∫

For example, ∫M(Ric(M))m

cm(ωWP)

VolωWP(M)

are all rational numbers.

Some remarks

The moduli space is often non-compact.

If the moduli space were compact, then the Chern-Weilforms define Chern classes, and the theorem follows.If the growth of the Chern-Weil forms and the connectionwere mild at infinity, the theorem follows from a theoremof Mumford.

Some remarks

The moduli space is often non-compact.If the moduli space were compact, then the Chern-Weilforms define Chern classes, and the theorem follows.

If the growth of the Chern-Weil forms and the connectionwere mild at infinity, the theorem follows from a theoremof Mumford.

Some remarks

The moduli space is often non-compact.If the moduli space were compact, then the Chern-Weilforms define Chern classes, and the theorem follows.If the growth of the Chern-Weil forms and the connectionwere mild at infinity, the theorem follows from a theoremof Mumford.

Important! We realized that

TheoremLet E → M be a Hodge bundle over the moduli space of apolarized Kähler manifold. Let R be the curvature tensor withrespect to the Hodge bundle. Then∫

The above result implies that the Chern-Weil forms areintegrable. But it is not that hard to prove the integrabilitycondition directly.

We first establish the following well-knownPropositionLet M any quasiprojective manifold. Then there is a Kählermetric on M such that

...1 It is complete;

...2 Its Ricci curvature has a lower bound; and

...3 it is of finite volume.The metric is called the Poincaré metric.

The above result implies that the Chern-Weil forms areintegrable. But it is not that hard to prove the integrabilitycondition directly. We first establish the following well-knownPropositionLet M any quasiprojective manifold. Then there is a Kählermetric on M such that

...3 it is of finite volume.

The metric is called the Poincaré metric.

Some big theoremsTheorem

...1 The moduli space of a polarized Kähler manifold is acomplex variety (Mumford);

...2 It is actually a quasi-projective variety (Viehweg);

...3 the infinity can be made divisors of normal crossings(Hironaka)

The construction of the Poincaré metric

...1 1d case, on the punctured disk, we have

ωP =dz ⊗ dz

|z|2(− log |z|)2

...2 high dimensional case: on (∆m)∗ ×∆n, we have

ωP =m∑

dzi ⊗ dzi

|zi|2(− log |zi|)2+

m+n∑i=m+1

dzi ⊗ dzi.

...3 on a quasi projective manifold: partition of unity.

ωP =dz ⊗ dz

|z|2(− log |z|)2

ωP =m∑

dzi ⊗ dzi

|zi|2(− log |zi|)2+

m+n∑i=m+1

dzi ⊗ dzi.

ωP =dz ⊗ dz

|z|2(− log |z|)2

ωP =m∑

dzi ⊗ dzi

|zi|2(− log |zi|)2+

m+n∑i=m+1

dzi ⊗ dzi.

The Schwarz Lemma of Yau:TheoremLet M be a Kähler manifold with two Kähler metrics ω1, ω2.Assume that

...1 Ric(ω1) ≥ −k1;

...2 the holomorphic sectional curvature of ω2 has a negativeupper bound −k2;

...3 with respect to ω1, M is complete.Then ω2 ≤ (k1/k2)ω1.

p Using the Schwartz lemma, we can prove that

ωH ≤ CωP,

where ωP is the Hodge metric.

p It is an algebraic fact that f(R) ≤ CωmH . The

integrability is proved.

p Using the Schwartz lemma, we can prove that

ωH ≤ CωP,

where ωP is the Hodge metric.p It is an algebraic fact that f(R) ≤ CωmH . The

integrability is proved.

Previous results

For the first Chern class of the Weil-Petersson metric, theresult is by Sun-L (CMP, 06);

For general moduli space, if the dimension is 1, the resultis by Peters, Zucker, Mumford (independently).For general moduli space, in the case of first Chern class,the result is by J. Kollár.

Previous results

For the first Chern class of the Weil-Petersson metric, theresult is by Sun-L (CMP, 06);For general moduli space, if the dimension is 1, the resultis by Peters, Zucker, Mumford (independently).

For general moduli space, in the case of first Chern class,the result is by J. Kollár.

Previous results

For the first Chern class of the Weil-Petersson metric, theresult is by Sun-L (CMP, 06);For general moduli space, if the dimension is 1, the resultis by Peters, Zucker, Mumford (independently).For general moduli space, in the case of first Chern class,the result is by J. Kollár.

Our result is true for the Weil-Petersson metric on Calabi-Yaumoduli and the Chern-Weil forms on general moduli spaces.We proved the theorems of this kind to the full generality, withthe introduction of new techniques.

The idea of MumfordLet M be the compactification of M. We suppose E can beextended to M. Let Γ0 be a smooth connection of theextended bundle. Let ρ be a cut-off function. We can comparethe two integrals:∫

Mρf(R) and

∫Mρf(R0)

where R0 = ∂Γ0 is the curvature of Γ0.

When ρ→ 1,∫M ρf(R) goes to

∫M f(R), and

∫M ρf(R0) goes

to∫M f(R0), which is an integer by the fact that the

Chern-Weil form defines a rational class.

The idea of MumfordLet M be the compactification of M. We suppose E can beextended to M. Let Γ0 be a smooth connection of theextended bundle. Let ρ be a cut-off function. We can comparethe two integrals:∫

Mρf(R) and

∫Mρf(R0)

where R0 = ∂Γ0 is the curvature of Γ0.

When ρ→ 1,∫M ρf(R) goes to

∫M f(R), and

∫M ρf(R0) goes

to∫M f(R0), which is an integer by the fact that the

Chern-Weil form defines a rational class.

Let f be the polarization of f. That is,

f(A, · · · ,A) = f(A)