A Global Torelli Theorem for Calabi-Yau Manifoldsscgas/scgas-2012/Talks/Liu.pdf · 2020-02-02 · Main topics I will present the proof of a global Torelli theorem for Calabi-Yau manifolds

A Global Torelli Theorem for Calabi-Yau Manifolds

Kefeng Liu

In Honor of Peter, A Great Geometer of Our Time

January 14, 2012

Kefeng Liu A Global Torelli Theorem for Calabi-Yau Manifolds

Happy Birthday! Peter!

The first time I met Peter was 20 years ago in 1992. For morethan 20 years he has helped and influenced me in many aspects ofmy life and career.

Peter is forever young! Today he still looks like 20 years ago whenhe was 40 years old!


Main topics

I will present the proof of a global Torelli theorem for Calabi-Yaumanifolds. This is joint work with Feng Guan and Andrey Todorov,with many ideas and helps from Yau.

Plan of the lecture

Introduction

The period map

Local geometry of Teichmuller space

Holomorphic affine structure on Teichmuller space

Holomorphic affine embedding of Teichmuller space in CN

Proof of the global Torelli theorem.

The main idea is to patch the local properties of variation ofHodge structures and deformation of Calabi-Yau manifolds byusing holomorphic affine structure.


Introduction

Torelli problem, one of major topics in algebraic geometry, has verylong history, starting from 1914. For a compact Kahler manifoldM, it asks whether the Hodge structure on its cohomology groups

Hr (M,C) =⊕

p+q=r

Hp,q(M)

can determine the complex structure of M.

Andreotti and Weil proved global Torelli for Riemann surfaces.

Shafarevich and Piatetski-Shapiro, Looijenga, Burns-Rapoportproved global Torelli theorem for K3 surfaces, a conjecture ofWeil.

Voisin and Looijenga proved global Torelli theorem for cubicfourfolds. Verbitsky recently proved a global Torelli theoremfor hyperKahler manifolds.

There are many works on local, generic Torelli theorems.Kefeng Liu A Global Torelli Theorem for Calabi-Yau Manifolds

Polarized and marked Calabi-Yau

Although our method works for more general cases, we will focuson Calabi-Yau case.

A compact projective manifold M of complex dimension n ≥ 3is called Calabi-Yau, if it has a trivial canonical bundle andsatisfies H i (M,OM) = 0 for 0 < i < n.

A pair (M, L) consisting of a Calabi-Yau manifold M ofcomplex dimension n and an ample line bundle L over M iscalled a polarized Calabi-Yau manifold.

The triple (M, L, γ1, · · · , γhn) is called a polarized andmarked Calabi-Yau manifold.

Here γ1, · · · , γhn be a basis of the integral homology groupmodulo torsion, Hn(M,Z)/Tor identified to a lattice Λ.


Primitive cohomology

Let M be a Calabi-Yau of complex dimension n. Still use L todenote the first Chern class of L. It defines a map

L : Hn(M,Q) → Hn+2(M,Q)

given by A 7→ L ∧ A for any A ∈ Hn(M,Q). Denote byHnpr (M) = ker(L) the primitive cohomology group.

Let Hk,n−kpr (M) = Hk,n−k (M) ∩ Hn

pr (M,C) and denote its

dimension by hk,n−k .We have the induced Hodge decomposition

Hnpr (M,C) = Hn,0

pr (M)⊕ · · · ⊕ H0,npr (M).


Primitive cohomology

The Poincare bilinear form Q on Hnpr (M,Q) is defined by

Q(u, v) = (−1)n(n−1)

2

∫

M

u ∧ v

for any d -closed n-forms u, v on M.Q is non-degenerate and can be extended to Hn

pr (M,C) bilinearly,and satisfies the Hodge-Riemann relations

Q(Hk,n−kpr (M),H l ,n−l

pr (M))= 0 unless k + l = n,

and

(√−1)2k−n

Q (v , v) > 0 for v ∈ Hk,n−kpr (M) \ 0.


Hodge filtration

Let f k =∑n

i=k hi ,n−i , and

F k = F k(M) = Hn,0pr (M)⊕ · · · ⊕ Hk,n−k

pr (M)

from which we have the decreasing filtration

Hnpr (M,C) = F 0 ⊃ · · · ⊃ F n.

We know that

dimC F k = f k , (1)

Hnpr (M,C) = F k ⊕ F n−k+1

and

Hk,n−kpr (M) = F k ∩ F n−k .


Period domain

In term of the Hodge filtration F n ⊂ · · · ⊂ F 0 = Hnpr (M,C), the

Hodge-Riemann relations can be written as

Q(F k ,F n−k+1

)= 0 (2)

and

Q (Cv , v) > 0 if v 6= 0, (3)

where C is the Weil operator given by Cv =(√

−1)2k−n

v when

v ∈ Hk,n−kpr (M). The classifying space D for polarized Hodge

structures is the space of all such Hodge filtrations, the locallyhomogeneous space

D =F n ⊂ · · · ⊂ F 0 = Hn

pr (X ,C) | (1), (2) and (3) hold.


Teichmuller space

A basis of (Hn(M,Z)/Tor)/m(Hn(M,Z)/Tor) is called a level mstructure on the polarized Calabi-Yau manifold. We have thefollowing theorem due to Popp, Viehweg and Szendroi.

Theorem

Let m ≥ 3 and M be polarized Calabi-Yau manifold with level m

structure, then there exists a quasi-projective complex manifold Zm

with a versal family of Calabi-Yau manifolds,

XZm→ Zm,

containing M as a fiber, and polarized by an ample line bundle

LZmon XZm

.


Teichmuller space

We define T = TL(M) to be the universal cover of Zm, and thefamily U → T to be the pull-back family.

Theorem

The Teichmuller space T is a simply connected smooth complex

manifold, and the family U → T containing M as a fiber, is local

Kuranishi at each point of T .

Local Kuranishi follows from unobstructedness of the deformationof Calabi-Yau manifolds. It means any local deformation of Mp forany p ∈ T can be induced from the family U.


Period map

The period map from T to D is defined by assigning each pointp ∈ T the Hodge structure on Mp,

Φ : T → D

with Φ(p) = F n(Mp) ⊂ · · · ⊂ F 0(Mp). Griffiths showed that Φis a holomorphic map.The main theorem of my talk:

Theorem

The period map Φ constructed above is an embedding,

Φ : T → D.


Local deformation

For a point p ∈ T , we denote by (Mp, L) the correspondingpolarized and marked Calabi-Yau manifold as the fiber over p.Let ωp ∈ L denote the Kahler form of the Calabi-Yau metric byYau’s theorem.Let (U, z1, · · · , zn) be the local coordinate chart, andΩ = fdz1 ∧ · · · ∧ dzn be a smooth (n, 0)-form on M,φ =

∑i φ

i ∂∂zi

∈ A0,1(M,T 1,0M) be a Beltrami differential. Wedefine

φyΩ =∑

i

(−1)i−1f φi ∧ dz1 ∧ · · · ∧ dzi ∧ · · · ∧ dzn.


Local deformation

The following lemma follows since Ωp is parallel with respect tothe Calabi-Yau metric. It also implies the local Torelli theorem forCalabi-Yau manifolds.

Lemma

Let Ωp be a nowhere vanishing holomorphic (n, 0)-form on Mp

such that

(√−1

2

)n

(−1)n(n−1)

2 Ωp ∧ Ωp = ωnp . (4)

Then the map ι : A0,1(M,T 1,0M

)→ An−1,1(M) given by

ι(φ) = φyΩp is an isometry with respect to the natural Hermitian

inner product on both spaces induced by ωp. Furthermore, ιpreserves the Hodge decomposition.


Kuranishi gauge, BTT

Let φ1, · · · , φN ∈ H0,1(Mp,T

1,0Mp

)be a basis of harmonic Beltrami

differemtials. Then there is a unique power series

φ(τ) =N∑

i=1

τiφi +∑

|I |≥2

τ IφI (5)

which converges for |τ | small. Here φI ∈ A0,1(Mp,T

1,0Mp

), and

∂Mpφ(τ) =

1

2[φ(τ), φ(τ)],

∂∗Mpφ(τ) = 0,

φIyΩp = ∂MpψI ,

(6)

for each |I | ≥ 2 where ψI ∈ An−2,1(Mp) are smooth(n − 2, 1)-forms.


Canonical family of holomorphic (n, 0)-forms

We fix on Mp a nowhere vanishing holomorphic (n, 0)-form Ωp andan orthonormal basis φiNi=1 of H1(Mp,T

1,0Mp). Let φ(τ) be theBeltrami differentials defining a local deformation of Mp which wedenote by Mτ .Locally write Ωp = dz1 ∧ · · · ∧ dzn. Introduce a smooth form onMτ ,

Ωcp(τ) = (dz1 + φ(dz1)) ∧ · · · ∧ (dzn + φ(dzn)) .

Here φ(dzi) = φi ∈ A0,1(M). Todorov proved that for τ small,Ωcp(τ) is a well-defined nowhere vanishing holomorphic (n, 0)-form

on Mτ depending on τ holomorphically. It can be proved by directcomputation to show dΩc

p(τ) = 0.


Local holomorphic flat coordinates

The following expansion in cohomology follows easily. We will useτ1, · · · , τN as the local holomorphic affine flat coordinates.

Theorem

We have the following expansion for |τ | small,

[Ωcp(τ)] = [Ωp] +

N∑

i=1

τi [φiyΩp] + A(τ), (7)

where A(τ) = O(|τ |2) ∈⊕nk=2 H

n−k,k (M) denotes terms of order

at least 2 in τ .

This expansion can also be derived from the local Torelli theoremfor Calabi-Yau manifold and the Griffiths transversality, aswell-developed in algebraic geometry.


Affine manifolds

Let M be a differentiable manifold of real dimension n, if there is acoordinate cover Ui , φi ; i ∈ I of M satisfying that,φik = φi φ−1

kis a real affine transformation on Rn, whenever

Ui ∩ Uk is not empty, then we say that Ui , φi ; i ∈ I is a realaffine coordinate cover and defines a real affine structure on M.

For the definition of holomorphic affine manifold we just replacethe ”real” by ”holomorphic” and ”Rn” by ”Cn” in the definition ofreal affine manifold.


Holomorphic flat connection

A linear connection ∇ on a complex manifold is called aholomorphic linear connection if the following two more conditionsare satisfied,

(∇YX )1,0 = ∇YX1,0;

If U and W are complex holomorphic vector fields defined onan open set O, then ∇WU is also holomorphic.

A holomorphic linear connection is called a holomorphic flatconnection if the following two additional conditions are satisfied,

∇XY −∇YX = [X ,Y ];

∇X∇Y −∇Y∇X −∇[X ,Y ] = 0.

This means that a holomorphic flat connection is a holomorphiclinear connection which is torsion-free and has zero curvature.


Characterization of holomorphic affine manifolds

The following theorem is due to Matsushima, Vitter. Its realanalogue was due to Auslander in 50s.

Theorem

Let M be a complex manifold, then there is a one-to-one

correspondence between the set of all holomorphic affine structures

on M and the set of all holomorphic flat torsion-free connections

on M. The geodesics of a holomorphic flat connection are straight

lines in each coordinate chart of the holomorphic affine coordinate

cover.


Holomorphic affine map

Let f : M → M ′ be a holomorphic affine map betweenholomorphic affine manifolds M and M ′. The following theorem isessentially due to Kobayashi-Nomizu.

Theorem

Let M be a connected, simply connected holomorphic affine

manifold and M ′ be a complete holomorphic affine manifold. Then

every holomorphic affine map fU of a connected open subset U of

M into M ′ can be uniquely extended to a holomorphic affine map

f of M into M ′.

Note that an affine map maps geodesics to geodesics, so straightlines to straight lines.


Hodge basis

Let e = e0, · · · , em be a basis of F 0 in the given Hodge filtration.We call e a Hodge basis adapted to this Hodge filtration, if

SpanCe0, · · · , emk = F n−k ,

where mk = f n−k − 1, for each 0 ≤ k ≤ n.

A square matrix T = [Tα,β], with each Tα,β a submatrix, is calledblock upper triangular if Tα,β is zero matrix whenever α > β. Toclarify the notations, we will use Tij to denote the entries of thematrix T . We use such matrix to transform Hodge basis to Hodgebasis.


Hodge basis

The following is a linear algebra lemma.

Lemma

We fix a base point p ∈ T and a Hodge basis c0(p), · · · , cm(p)of the Hodge filtration of Mp. Then there is an open neighborhood

Up of p, such that for any q ∈ Up, there exists a block upper

triangular matrix σ(q) such that the basis

c0(q)...

cm(q)

= σ(q)

c0(p)...

cm(p)

(8)

is a Hodge basis of the Hodge filtration of Mq.

It follows from Gauss eliminations or from Griffiths transversality.


Kuranishi coordinate cover

For each p ∈ T , we denote by Cp the set consisting of all of theorthonormal bases for H0,1(Mp,T

1,0Mp). Then for each pair(p,Ψ), where p ∈ T and Ψ ∈ Cp, we call the following coordinatechart

Up,Ψ = (Up, τ1, · · · τN) = (Up, τ)

a holomorphic affine flat coordinate chart around p. It gives us acoordinate cover of T , which we call the Kuranishi coordinatecover.


Holomorphic affine structure on the Teichmuller space

Our first key observation is that the Kuranishi coordinate covergives us a natural holomorphic affine coordinate cover of T ,therefore a holomorphic affine structure on T .

Theorem

The Kuranishi coordinate cover on T is a holomorphic affine

coordinate cover, thus defines a global holomorphic affine structure

on T .

The proof of this theorem uses the Calabi-Yau property in a crucialway. It is so simple that we can explain the detail of the argument.

Holomorphic affine structure should be a common global feature inTorelli problem.



The key to prove the above theorem is the following

Lemma

Let p, q ∈ T be two points. If q ∈ Up, then the transition map

between (Up, τ) and (Uq, t) is a holomorphic affine map.

We take Φp = φ1, · · · , φN and Ψq = ψ1, · · · , ψN to be theorthonormal bases of H0,1(Mp,T

1,0Mp) and H0,1(Mq ,T1,0Mq)

respectively.Let Ωp and Ωq be respectively the holomorphic (n, 0)-forms on Mp

and Mq. Then φ1yΩp , · · · , φNyΩp and ψ1yΩq , · · · , ψNyΩqare respectively the orthonormal bases for Hn−1,1(Mp) andHn−1,1(Mq).



Write

η0 = [Ωp], ηi = [φiyΩp] for 1 ≤ i ≤ N;

α0 = [Ωq], αi = [ψjyΩq] for 1 ≤ j ≤ N.

We complete them to Hodge bases for Mp and Mq respectively,

η = (η0, η1, · · · , ηN , · · · , ηm)T for Mp;

α = (α0, α1, · · · , αN , · · · , αm)T for Mq.

For any point r ∈ Up ∩ Uq, let us compute the transition mapbetween the holomorphic affine flat coordinates at r ,(τ1(r), · · · , τN(r)) and (t1(r), · · · , tN(r)).



Let [Ωr ] = [Ωcp(τ(r))] ∈ Hn,0(Mr ), where [Ωc

p(τ)] is the canonicalsection of the holomorphic (n, 0)-classes around p. Then we havethe following identities:

[Ωr ] = η0 +N∑

i=1

τi(r)ηi +∑

k>N

fkηk ; (9)

[Ωr ] = λ

(α0 +

N∑

i=1

ti(r)αi +∑

k>N

gkαk

). (10)

Here each fk(r) is the coefficient of ηk in the decomposition of[Ωr ] according to the Hodge decomposition on Mp, and each gk(r)is the coefficient of αk in the decomposition of λ−1[Ωr ] accordingto the Hodge decomposition on Mq, for N + 1 ≤ k ≤ m.



We know that there is a Hodge basis corresponding to the Hodgestructure of Mq,

C (q) = (c0(q), · · · , cN(q), · · · , cm(q)),

such that ci (q) =∑j

σijηj , and the matrix σ = [σα,β ] is block

upper triangular and nonsingular. Since we have two bases of thesame Hodge filtration on Mq, C (q) and α. They are related by anonsingular block diagonal transition matrix of the form

A =

A0,0 0 · · · 00 A1,1 · · · 0· · · · · · · · · · · ·0 0 · · · An,n

where each Aα,α is an invertible hn−α,α × hn−α,α matrix, for0 ≤ α ≤ n.



Thus we have

α =A · C (q) and C (q) = σ · η,

from which we get the transition matrix between the basis α andbasis η, α = Aση.It is clear that Aσ is still a nonsingular block upper triangularmatrix of the form

A0,0 · σ0,0 ∗ · · · ∗0 A1,1 · σ1,1 · · · ∗· · · · · · · · · · · ·0 0 · · · An,n · σn,n

.



Let us denote Aσ by T = [Tα,β ]0≤α,β≤n. Note that each Tα,β isan hn−α,α × hn−β,β matrix for 0 ≤ α, β ≤ n, and each Tα,α isinvertible.In particular note that the 1× 1 matrix T 0,0 = [T00] and theN × N matrix T 1,1 in T are nonsingular, which are used in thecomputation of holomorphic affine flat coordinate transformationin the following.Then we project [Ωr ] to F n−1

p = Hn,0(Mp)⊕ Hn−1,1(Mp). LetPn−1p denote the projection from F 0

p to F n−1p .



From (9) and (10) we see that

η0 +

N∑

i=1

τiηi = λPn−1p (α0 +

N∑

i=1

αi ti +

m∑

k=N+1

gk(t)αk)

= λPn−1p (

m∑

j=0

T0jηj +

N∑

i=1

ti

m∑

j=0

Tijηj +

m∑

k=N+1

gk(t)

m∑

j=0

Tkjη

= λ(

N∑

j=0

T0jηj +

N∑

i=1

ti

N∑

j=0

Tijηj) (11)

= λT00η0 +N∑

j=1

(λT0j +N∑

i=1

λTij ti)ηj .

Here for brevity we have dropped the notation r in the coordinatesτi (r) and ti (r) in the above formulas.



By comparing the coefficients of the basis η1, · · · , ηN on bothsides of the above identity, we get 1 = T00λ and

τj = λT0j +N∑i=1

λTij ti . Therefore for 1 ≤ j ≤ N, we have the

identity,

τj = T−100 T0j +

N∑

i=1

T−100 Tij ti .

In this identity, one notes that the transition matrix T , whiledepending on p and q, is independent of r . Thus we have provedthat the coordinate transformation is a holomorphic affinetransformation.



Let (Up, τ) and (Uq, t) be two holomorphic affine flat coordinatecharts in the Kuranishi coordinate cover. If p = q, it is easy to seethe transition map between these two coordinate charts is aholomorphic affine map by basis change matrix.If p 6= q, then we use a smooth curve γ(s) to connect p = γ(0)and q = γ(1). Then we can easily choose

0 = s0 < s1 < · · · < sk−1 < sk = 1,

such that γ(sl+1) ∈ Uγ(sl ) and the transition maps φl ,l+1 betweenthe holomorphic affine flat coordinates in Uγ(sl ) and Uγ(sl+1) areholomorphic affine maps.



Then the transition map between the holomorphic affine flatcoordinates in Up and Uq, which is the following compositions ofholomorphic affine maps,

φpq = φ0,1 · · · φk−1,k

is also a holomorphic affine map, whenever Up ∩ Uq is not empty.This completes the proof of the existence of holomorphic affine flatstructure on T .


Holomorphic affine embedding of T in CN

Take a base point p ∈ T , and choose a holomorphic flatcoordinate chart (Up, τ1, · · · , τN) around p. Define a localholomorphic affine embedding

ρUp: Up → CN ∼= Hn−1,1(Mp)

by letting ρUp(q) = (τ1(q), · · · , τN(q)) for any q ∈ Up. Here recall

the identification Hn−1,1(Mp) ∼= CN by using the orthonormalbasis [φ1yΩp], · · · , [φNyΩp] of Hn−1,1(Mp).We can extend ρUp

to a holomorphic affine map by the theorem ofKobayashi-Nomizu,

ρp : T → CN ,

such that when restricted to Up, one has ρp|Up= ρUp

.



We have the following theorem.

Theorem

The holomorphic affine map ρp : T → CN is an embedding.

The proof is to first construct global holomorphic affine flatcoordinates on T centered at p, then we show that the ρp isactually given by the coordinate functions with ρp(p) = 0.

We have two proofs of this theorem, one by using the holomorphicflat connection to construct global coordinate functions, another isby using elementary topological argument. I will present thesimpler one.


Global holomorphic affine coordinates on T

This proof is basically the same as the proof that flat vector bundleon a simply connected manifold is trivial.

Let (Uα, φα) : α ∈ I be the Kuranishi coordinate cover, whereφα : Uα → CN denotes the coordinate map.Denote by Aff(CN) the group of holomorphic affine transformationsfrom CN to CN . Then using the fact that the Kuranishi coordinatecover on T is a holomorphic affine coordinate cover, we get thatthe transition map φα φ−1

β ∈ Aff(CN) for each α and β.



For a fixed point p ∈ T and the holomorphic affine flat coordinatechart τ1, · · · , τN around p, it is not hard to see that in the groupAff(CN), we can choose an Aα ∈ Aff(CN) for each coordinatechart (Uα, φα), such that

Aα φα = Aβ φβ whenever Uα ∩ Uβ is not empty;

Ap = Id on Up.

The obstruction for this is in Hom (π1(T ), Aff(CN)) whichvanishes, because T is simply connected.



This implies that we can choose a new coordinate cover,

(Uα, ψα = Aα φα) : α ∈ I,

which satisfies that

ψα ψ−1β = Id for Uα ∩ Uβ not empty.

This gives us the global holomorphic affine flat coordinates whichagree with τ1, · · · , τN on Up. We denote the global holomorphicaffine coordinates centered at p by t1, · · · , tN.



By definition we know that when restricted to Up, we have

ρi(q) = τi(q) = ti(q) for q ∈ Up (12)

where we write ρp(q) = (ρ1(q), · · · , ρN(q)) ∈ CN .Note that for each i , the coordinate function ti and ρi are bothglobally defined holomorphic functions on T . This implies that

ρi(q) = ti(q) for q ∈ T .

Therefore the holomorphic affine mapρp : T → CN ∼= Hn−1,1(Mp), actually given by the globalholomorphic affine flat coordinates on T , is an embedding.


Proof of the global Torelli theorem

Now we are ready to prove our main theorem.

Theorem

The period map Φ : T → D is an embedding.

Recall that for arbitrary p ∈ T , the previous theorem gives us aglobal holomorphic affine flat coordinate system t1, · · · , tN in Tcentered at p, and an embedding ρp : T → CN such that for anyq ∈ T ,

ρp(q) = (t1(q), · · · , tN(q)).

In particular we have ρp(q) = 0 if and only if q = p.


Proof of global Torelli for Calabi-Yau

For any q ∈ T different form p, we connect p and q by a smoothcurve γ(s) with γ(0) = p and γ(1) = q. We can choose

0 = s0 < s1 < · · · < sk = 1

such that, γ(sζ+1) ∈ Uγ(sζ), where Uγ(sζ) is the holomorphic affineflat coordinate chart at γ(sζ). We know that there exists anonsingular block upper triangular matrices Tζ , such that

C (sζ+1) = Tζ · C (sζ),

where C (sζ) = (c0(sζ), · · · , cm(sζ)) is a Hodge basis of theCalabi-Yau manifold Mγ(sζ) at the point γ(sζ), for 0 ≤ ζ ≤ k .



The matrices Tζ all have the same pattern adapted to the Hodgefiltration, and we have C (q) = T · C (p) where C (p) = C (s0),C (q) = C (sk) with

T = [Tα,β]0≤α,β≤n =k−1∏

ζ=0

Tζ .

Here T is a nonsingular block upper triangular matrix of the samepattern as Tζ , that is, T

α,β is an hn−α,α × hn−β,β matrix for each0 ≤ α, β ≤ n, and Tα,α is nonsingular for each α.



Comparing the first term of the basis we have

c0(q) = T00c0(p) +N∑

j=1

T0jcj(p) +m∑

l=N+1

T0lcl (p), (13)

where c0(q) is a generator of Hn,0(Mq). Here recall that N is thedimension of Hn−1,1(Mp).



Let t ′1, · · · , t ′N be the global holomorphic affine flat coordinatesystem centered at q. Then we also get the following holomorphicaffine transformation,

tj = T−100 T0j +

N∑

i=1

T−100 Tijt

′i (14)

for 1 ≤ j ≤ N. This is the transition map of the two globalholomorphic affine flat coordinate systems t1, · · · , tN andt ′1, · · · , t ′N. The coordinates of q is given by

tj(q) = T−100 T0j ,

since t ′i(q) = 0 for each 1 ≤ i ≤ N.



Suppose there exists a point q ∈ T which is different from p, suchthat Hn,0(Mq) = Hn,0(Mp). Then (13) shows that T0j = 0, for1 ≤ j ≤ N. We get

tj(q) = T−100 T0j = 0 for 1 ≤ j ≤ N.

But by the definition of global holomorphic affine flat coordinatechart t1, · · · , tN, this means p = q, which contradicts to that qis different from p. This completes the proof of the theorem.


Thank You!


Documents

A Global Torelli Theorem for Calabi-Yau Manifoldsscgas/scgas-2012/Talks/Liu.pdf · 2020-02-02 · Main topics I will present the proof of a global Torelli theorem for Calabi-Yau manifolds