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 ∫

Mf(R)

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 ∫

Mf(R)



For example, ∫M(Ric(M))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∫

Mf(R)



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∑

i=1

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∑

i=1

dzi ⊗ dzi

|zi|2(− log |zi|)2+

m+n∑i=m+1

dzi ⊗ dzi.





ωP =dz ⊗ dz

|z|2(− log |z|)2


ωP =m∑

i=1

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)

and

f(· · · ,Ai, · · · ,Aj, · · · ) = f(· · · ,Aj, · · · ,Ai, · · · )

and

f(TA1T−1, · · · ,TAjT−1, · · · ) = f(A1, · · · ,Aj, · · · ).

f is linear with respect to all the components.For example, if f = tr(A2), then f = 1

2tr(AB + BA).In general,

all invariant polynomials have a unique polarization.



f(A, · · · ,A) = f(A)

and

f(· · · ,Ai, · · · ,Aj, · · · ) = f(· · · ,Aj, · · · ,Ai, · · · )

and

f(TA1T−1, · · · ,TAjT−1, · · · ) = f(A1, · · · ,Aj, · · · ).






f(A, · · · ,A) = f(A)

and

f(· · · ,Ai, · · · ,Aj, · · · ) = f(· · · ,Aj, · · · ,Ai, · · · )

and

f(TA1T−1, · · · ,TAjT−1, · · · ) = f(A1, · · · ,Aj, · · · ).






f(A, · · · ,A) = f(A)

and

f(· · · ,Ai, · · · ,Aj, · · · ) = f(· · · ,Aj, · · · ,Ai, · · · )

and

f(TA1T−1, · · · ,TAjT−1, · · · ) = f(A1, · · · ,Aj, · · · ).

f is linear with respect to all the components.

For example, if f = tr(A2), then f = 12tr(AB + BA).In general,




f(A, · · · ,A) = f(A)

and

f(· · · ,Ai, · · · ,Aj, · · · ) = f(· · · ,Aj, · · · ,Ai, · · · )

and

f(TA1T−1, · · · ,TAjT−1, · · · ) = f(A1, · · · ,Aj, · · · ).


2tr(AB + BA).

In general,all invariant polynomials have a unique polarization.



f(A, · · · ,A) = f(A)

and

f(· · · ,Ai, · · · ,Aj, · · · ) = f(· · · ,Aj, · · · ,Ai, · · · )

and

f(TA1T−1, · · · ,TAjT−1, · · · ) = f(A1, · · · ,Aj, · · · ).





Then we have the following identity:∫Mρ(f(R)− f(R0))

=∑

i

∫Mρf(R, · · · ,R,R − R0,R0, · · · ,R0)

= −∑

i

∫M∂ρ ∧ f(R, · · · ,R,Γ− Γ0,R0, · · · ,R0)

Apparently, if the growth of R,R0,Γ,Γ0 are mild, then theright hand side will go to zero.

If a connection whose growthis mild, Mumford called it “good”.


Then we have the following identity:∫Mρ(f(R)− f(R0))

=∑

i

∫Mρf(R, · · · ,R,R − R0,R0, · · · ,R0)

= −∑

i

∫M∂ρ ∧ f(R, · · · ,R,Γ− Γ0,R0, · · · ,R0)

Apparently, if the growth of R,R0,Γ,Γ0 are mild, then theright hand side will go to zero. If a connection whose growthis mild, Mumford called it “good”.


.Unfortunately,..

.. ..

.

.the Hodge metrics are not “good” in the sense of Mumford.

What do we need to do?Control ∂ρ;Control Γ,R;Control Γ0,R0.


.Unfortunately,..

.. ..

.


What do we need to do?Control ∂ρ;

Control Γ,R;Control Γ0,R0.


.Unfortunately,..

.. ..

.


What do we need to do?Control ∂ρ;Control Γ,R;

Control Γ0,R0.


.Unfortunately,..

.. ..

.




.Unfortunately,..

.. ..

.




The P-boundedness: How “good” is enough?DefinitionLet U be a neighborhood and let z1, · · · , zn be theholomorphic coordinate system. The divisor D = z1 = 0. Asmooth form on U\D is called P bounded (refer to PoincaréBounded), if it is bounded under the frame

d log z1/ log 1

|z1|, d log z1/ log 1

|z1|, dzj, dzj

for j > 1.

For example, a one form η = aidzi is P bounded iff

|a1| ≤C

r1 log 1/r1, |ak| ≤ C, for k > 1


The P-boundedness: How “good” is enough?DefinitionLet U be a neighborhood and let z1, · · · , zn be theholomorphic coordinate system. The divisor D = z1 = 0. Asmooth form on U\D is called P bounded (refer to PoincaréBounded), if it is bounded under the frame

d log z1/ log 1

|z1|, d log z1/ log 1

|z1|, dzj, dzj

for j > 1.For example, a one form η = aidzi is P bounded iff

|a1| ≤C

r1 log 1/r1, |ak| ≤ C, for k > 1


If η1, η2 are P bounded, then so is η1 ∧ η2. Furthermore, if η isan (n, n) form which is P-bounded, then

η ≤ Cn∏

i=1

1

|zi|2(log 1|zi|)

2

In particular,∫

U η is finite.It is standard to prove that one can choose a cut-off functionρ so that ∂ρ is P-bounded with the Lebesgue of supp (∂ρ)going to zero. For example, we can pick ρ = g(log 1/r).



η ≤ Cn∏

i=1

1

|zi|2(log 1|zi|)

2

In particular,∫

U η is finite.

It is standard to prove that one can choose a cut-off functionρ so that ∂ρ is P-bounded with the Lebesgue of supp (∂ρ)going to zero. For example, we can pick ρ = g(log 1/r).



η ≤ Cn∏

i=1

1

|zi|2(log 1|zi|)

2

In particular,∫




η ≤ Cn∏

i=1

1

|zi|2(log 1|zi|)

2

In particular,∫



We consider a neighborhood U at the infinity of the modulispace, where the divisor D can be written as z1 = 0. By thenilpotent orbit theorem of Schmid, we have the expansion ofthe Hodge metric

h =

(

log 1r1

)α1

. . . (log 1

r1

)αr

+ lower order terms

Without loss of generality, we assume that

α1 ≥ α2 · · · ≥ αr


We let

Λ =

(

log 1r1

)α1/2

. . . (log 1

r1

)αr/2

Then we have

h = Λh′Λ

Then we have h′ = I + lower order terms, and the connectionis .

.. ..

.

.∂hh−1 = ∂ΛΛ−1 + Λ∂h′(h′)−1Λ−1 + Λh′∂ΛΛ−1(h′)−1Λ−1


Go back to the previous expression

∂ρ ∧ f(R, · · · ,R,Γ− Γ0,R0, · · · ,R0)


Since Γ0,R0 are smooth, they are bounded.

However, even inthe above simplest case, the connection and hence thecurvature are not P bounded:

∂hh−1 = ∂ΛΛ−1 + Λ∂h′(h′)−1Λ−1 + Λh′∂ΛΛ−1(h′)−1Λ−1

But

Ad(Λ−1)(∂hh−1) = Λ−1∂Λ + ∂h′(h′)−1 + h′∂ΛΛ−1(h′)−1

is P bounded.In fact, this is a special case of the Cattani-Kaplan-Schmidtheorem.


Since Γ0,R0 are smooth, they are bounded. However, even inthe above simplest case, the connection and hence thecurvature are not P bounded:

∂hh−1 = ∂ΛΛ−1 + Λ∂h′(h′)−1Λ−1 + Λh′∂ΛΛ−1(h′)−1Λ−1

But

Ad(Λ−1)(∂hh−1) = Λ−1∂Λ + ∂h′(h′)−1 + h′∂ΛΛ−1(h′)−1




∂hh−1 = ∂ΛΛ−1 + Λ∂h′(h′)−1Λ−1 + Λh′∂ΛΛ−1(h′)−1Λ−1

But

Ad(Λ−1)(∂hh−1) = Λ−1∂Λ + ∂h′(h′)−1 + h′∂ΛΛ−1(h′)−1

is P bounded.

In fact, this is a special case of the Cattani-Kaplan-Schmidtheorem.



∂hh−1 = ∂ΛΛ−1 + Λ∂h′(h′)−1Λ−1 + Λh′∂ΛΛ−1(h′)−1Λ−1

But

Ad(Λ−1)(∂hh−1) = Λ−1∂Λ + ∂h′(h′)−1 + h′∂ΛΛ−1(h′)−1



The problem is that,when

Ad(Λ−1)(∂hh−1)

is P bounded, then ingeneral

Ad(Λ−1)(Γ0) and Ad(Λ−1)(R0)

are not bounded.

Λ =

(log 1

r1

)α1/2

. . . (log 1

r1

)αr/2


LetΓ0 = (aij)

ThenAd(Λ−1)(Γ0) = (aij(log 1/r1)αi−αj),

and they are bounded if and only if Γ0 is of lower triangular.

The technical heart of the proof is that

We are able to find such a connection.


LetΓ0 = (aij)

ThenAd(Λ−1)(Γ0) = (aij(log 1/r1)αi−αj),

and they are bounded if and only if Γ0 is of lower triangular.The technical heart of the proof is that

We are able to find such a connection.


Let Ωα be the frame defined by the nilpotent theorem and letAαβ be the transition functions. We proved that theconnection

Γα =∑

ψγ∂AαγA−1αγ

is the Correct connection! Or roughly speaking, they are all oflower triangular!


The proof is very technical and uses the full power of theSL2-orbit theorem of several variables.


The key technical lemma of this section is the following:LemmaLet U′ = Uγ be an open set such that U′ ∩ C = ∅. LetA = Aαγ. Then on U′ ∩ C = ∅,

Ad((e−1)t)Ad(A−1C )(∂AA−1)

is Poincaré bounded.


Sketch of the Proof. Let Ω′ = Ωγ be the local frame of U′

defined by the nilpotent orbit theorem. Let C′ be a fixed coneof U′ and let Ω′

C′ be the frame of the cone. We assume thatC ∩ C′ = ∅. We just need to prove the assertion of the lemmaon C ∩ C′ because as C′ is running over all the cones, thewhole U′ ∩ C will be covered.Let e′ be the matrix under the frame Ω′

C′ . Let AC′ be theconstant matrix defined as

Ω′ = Ω′C′At

C′ .

Then we haveΩC = Ω′

C′AtC′At(A−1

C )t.


We let B = A−1C AAC′ and let h′ be the metric matrix of Ω′

C′ .Then ΩC = Ω′

C′Bt. Thus

h = ΩtCΩC = Bh′Bt. (1)

It follows that

∂hh−1 = ∂BB−1 + Ad(B)(∂h′(h′)−1). (2)

Since AC,AC′ are constant matrices, we have

∂BB−1 = Ad(A−1C )(∂AA−1).


In order to prove the lemma, we only need to prove that

Ad((e−1)t)Ad(B)∂h′(h′)−1 = D(Ad(((e′)−1)t))(∂h′(h′)−1)D−1

is Poincaré bounded, where D = (e−1)tB(e′)t . Thus we onlyneed to prove that D and D−1 are bounded.


To prove the claim in the last slide, we observe that if

h = etke,

and ifh′ = (e′)tk′e′,

then by (1)k = Dk′Dt.

As a consequence of the SL2 orbit theorem ofCattani-Kaplan-Schmid, Since both k and k′ are positivedefinite, D and D−1 must be bounded.


Physics Background

In the paper of Ashok-Douglas the index of all supersymmetricvacua was given. The index is given by

Ivac(L ≤ Lmax) = const.∫M×H

det(−RWP − ωWP),

where M is the Calabi-Yau moduli and H is the moduli spaceof elliptic curves.


In the paper of Douglas-Shiffman-Zelditch, the followingstrengthened result of the above was given.TheoremLet K be a compact subset of M with piecewise smoothboundary. Then

IndχK(L) = const.(L2m)

[∫K

cm(T∗(M)⊗F3) + O(L−1/2)

].


By our results, we haveTheoremThe indices Ivac and IndχK are all finite. Moreover, IndχK isbounded from above uniformly with respect to K. They are allbounded, up to an absolute constant, by the Hodge volume ofthe Calabi-Yau moduli.


In Layman words, if string theory is true, the the number offeasible Universes is finite.


Thank you!


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,

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