calabi-yau equation anna fino calabi-yau equation on ...the kt manifold ak structure on the kt...

Calabi-Yau equation

Anna Fino

BackgroundSymplectic CY problem

Uniqueness of solutions

Existence of solutions

The CY equation onthe KT manifoldAK structure on the KTmanifold

Tosatti and Weinkove result

CY equation on the KTmanifold II

Main resultClassification of T 2 -bundlesover T2

Classification of theinvariant AK structures

Solutions of the generalizedMA equation

References

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Calabi-Yau equation on symplecticT 2-bundles over T 2

A joint work with Y. Y. Li, S. Salamon and L. Vezzoni

EMS-RSME Joint Mathematical Weekend, Bilbao – 7-9October 2011

Anna FinoDipartimento di Matematica

Università di Torino

Calabi-Yau equation

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References

2

1 BackgroundSymplectic CY problemUniqueness of solutionsExistence of solutions

2 The CY equation on the KT manifoldAK structure on the KT manifoldTosatti and Weinkove resultCY equation on the KT manifold II

3 Main resultClassification of T 2-bundles over T2Classification of the invariant AK structuresSolutions of the generalized MA equation

4 References

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References

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The Calabi-Yau problem

Theorem (Yau, Symplectic version)

Let (M2n, J,Ω) be a compact Kähler manifold and let σ be avolume form satisfying

∫M Ω

n =∫

M σ. Then there exists aunique Kähler form Ω̃ with [Ω̃] = [Ω] such that

Ω̃n = σ ←− CY Equation

Yau’s original proof of the existence makes use of a continuitymethod between the prescribed volume form σ and the naturalvolume form Ωn.• The proof of openness is by the implicit function theorem.• The closedness part is obtained by a priori estimates.

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Donaldson introduced the symplectic version of the Calabi-Yauequation for almost-Kähler manifolds.

Definition

An almost-Kähler (AK) manifold is a symplectic manifold (M,Ω)together with an almost cx structure J satisfying:

(a) Ω(X , JX ) > 0, ∀X 6= 0,(b) Ω(JX , JY ) = Ω(X ,Y ), ∀X ,Y .

• Associated to the AK structure (Ω, J) there is g(·, ·) = Ω(·, J·).

• If (a) holds, but not necessarily (b), then we say that Ω tamesJ and there exists

g(X ,Y ) =12

(Ω(X , JY )− Ω(JX ,Y )).

Calabi-Yau equation

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Symplectic CY problem

Let (M, J,Ω,g) be a 2n-dimensional compact AK manifold witha volume form σ = eF Ωn satisfying

∫M e

F Ωn =∫

M Ωn. Then

CY equation←→ (Ω + dα)n = eF Ωn (∗)

where Ω + dα is assumed to be a positive (1,1)-form.

• (∗) is elliptic for n = 2;• (∗) is overdetermined for n > 2.

Calabi-Yau equation

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Proposition (Donaldson)

If n = 2 the solutions to the Calabi-Yau problem are unique.

Proof: Let Ω1 and Ω2 be two solutions to the CY problem.Then {

Ω21 = Ω22 ,

Ω2 = Ω1 + dα=⇒ dα ∧ dα + 2Ω1 ∧ dα = 0 .

Consider Ω̄ = Ω1 + Ω2. Ω̄ is a symplectic form.

Ω̄ ∧ dα = 0 =⇒ dα is self-dual and then dα = 0 .

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Related conjectures

Conjecture (Donaldson)

Let M4 be compact with an almost cx structure J and a tamingsymplectic form Ω. Let σ be a normalized C∞ volume form onM4, i.e. s. t.

∫M4 σ =

∫M4 Ω

2.Then if Ω̃ is a AK form with [Ω̃] = [Ω] and solving the CYequation Ω̃2 = σ, there are C∞ a priori bounds on Ω̃ dependingon Ω, J, σ.

If this conjecture were to hold, by Donaldson, it would imply

Conjecture (Donaldson)

Let M4 compact with b+(M) = 1 and let J be an almost cxstructure. If there exists a symplectic form Ω on M4 taming Jthen there exists a symplectic form compatible with J.

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Let (M4,Ω, J,g) be AK manifold. Thn ∃! connection ∇C (thecanonical or Chern connection) such that

∇CJ = ∇CΩ = 0 , Tor1,1(∇C) = 0 .

ConsiderRi jk l = R

jik l

+ 4N rl jNirk.

Theorem (Tosatti,Weinkove,Yau)

If R(g, J) ≥ 0, i.e. Ri jk lXiX

jY k Y

l ≥ 0,∀X ,Y , then the firstDonaldson’s conjecture holds and the Calabi-Yau problem canbe solved for every normalized volume form on (M4,Ω, J,g).

Example (Tosatti,Weinkove,Yau)

If R(g, J) ≥ 0, then the first conjecture holds in any dimension2n. It can be applied to an infinitesimal deformation of theFubini-Study Kähler structure on CPn.

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References

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The CY equation on the Kodaira-Thurston manifold

Remark

The existence result by Tosatti-Weinkove-Yau cannot beapplied to the KT manifold M4 = (Γ\Nil3)× S1, where

Nil3 =

1 x z0 1 y

0 0 1

, x , y , z ∈ R

• M4 has a global invariant coframe {ei} such that

e1 = dy , e2 = dx , e3 = dt , e4 = dz − x dy .

We will denote simply Nil3 × R by (0,0,0,12).

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• M4 is the total space of a T 2-bundle over T2:

T 2 = S1 × S1 ↪→ Γ\Nil3 ×S1↓ πxyT2xy

and πxy is holomorphic with respect to an invariant complexstructure J̃.

• (M4, J1) has a holomorphic symplectic form given by

(e1 + ie2) ∧ (e3 + ie4).

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AK structure on the KT manifold

• M4 has the AK structure

Ω = e1 ∧ e4 + e2 ∧ e3, Je1 = e4, Je2 = e3, g =4∑

i=1

ei ⊗ ei .

• The symplectic form Ω is Lagrangian w.r. to the T 2-fibration

πxy : Γ\Nil3 × S1 −→ T2xy ,

i.e. Ω vanishes on the fibers.

• Λ+ = span < e12 + e34,e13 − e24,e14 + e23 >.

• b1(M4) = 3, b+(M4) = 2 and M4 does not admit any Kählerstructure.

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Tosatti and Weinkove solved the symplectic CY problem on theKT manifold (M4, J,Ω,g).

Theorem (Tosatti,Weinkove)

The CY equation on the KT manifold (M4, J,Ω,g) can besolved for every T 2-invariant volume form σ.

Argument of the proof:1 Writing σ = eF Ω2, for a C∞ T 2-invariant function F , then

by the normalization of σ one has∫

M4 eF Ω2 =

∫M4 Ω

2.

Every solution Ω̃ = Ω + dα of the CY problem satisfies thefollowing a priori bound on the metric g̃ associated to(Ω̃, J):

trg g̃ ≤ MinM4 ∆ F .

2 The continuity method gives the result.

Calabi-Yau equation

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References

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CY equation on the KT manifold II

Consider the Calabi-Yau equation (Ω + dα)2 = eF Ω2. Let

α = v e1 + vx e3 + vy e4 , v ∈ C∞(T2) .

Thendα = vxx e23 + vxy (e13 + e24) + vyy e14

and the CY equation becomes the Monge-Ampère equation

(1 + vxx )(1 + vyy )− v2xy = eF

Theorem (Li)

The Monge-Ampère equation on the standard torus T 2 has aunique solution.

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Goal: To generalize this argument to other AK structures onT 2-bundles over T2.

Definition (Thurston)

A geometric 4-manifold is a pair (X ,G) where X is a complete,simply-connected Riemannian 4-manifold, G is a group ofisometries acting transitively on X that contains a discretesubgroup Γ such that Γ\X has finite volume.

Let Nil4 = (0,13,0,12), Sol3 × R = (0,0,13,41).

Theorem (Ue)

Every orientable T 2-bundle over T2 is a geometric 4-manifold,where (X ,G) is one of the following

(R4,SO(4) nR4), (Nil3 × R,Nil3 × S1),(Nil4,Nil4), (Sol3 × R,Sol3 × R)

and it is infra-solvmanifold, i.e. a smooth quotient Γ\X coveredby a solvmanifold.

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Equivalently, every orientable T 2-bundle over T2 is a quotientΓ\X , where Γ is a discrete group containing a lattice Γ̃ of Xsuch that Γ̃\Γ is finite.

Definition

An AK structure (J,Ω,g) on an infra-solvmanifold M4 = Γ\X iscalled invariant if it induced by a left-invariant on X and it isΓ-invariant.

Proposition (-, Li, Salamon, Vezzoni)

On a 4-dimensional infra-solvmanifold (Γ\X , J,Ω,g) with aninvariant AK structure, the Tosatti-Weinkove-Yau conditionR(g, J) ≥ 0 is satisfied if and only if (J,Ω) is Kähler.

Calabi-Yau equation

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References

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The main result

Theorem (–, Li, Salamon, Vezzoni)

Let M4 = Γ\X be a T 2-bundle over a T2, with X = Nil3 × R orNil4 and suppose that M4 has an invariant Lagrangian AKstructure (J,Ω,g). Then for every normalized T 2-invariantvolume form σ = eF Ω2, F ∈ C∞(T2) the associated CYproblem has unique solution.

Layout of the proof:

• Use the classification of T 2-bundles of T2;• Classify in each case all the invariant Lagrangian almost

Kähler structures;• Rewrite the problem in terms of a generalized

Monge-Ampere equation;• Show that such an equation has a solution.

Calabi-Yau equation

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References

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For the non Lagrangian case:

Proposition (–, Li, Salamon, Vezzoni)

Let Ω be an invariant symplectic form on M4 = Γ\X andsuppose that Ω 6= 0 on the fibers of π : M4 −→ T2. Then ∃ aninvariant AK structure (J,Ω,g) on M4 such that π isJ-holomorphic.In this situation, for every normalized T 2-invariant σ = eF Ω2,F ∈ C∞(T2) the associated CY problem has a solution.

Remark

• The Lagrangian condition may or not apply in the case ofX = Nil3 × R, but is automatic when X = Nil4.• If X = Sol3 × R the corresponding T 2-bundle does not admitLagrangian fibration but we may apply previous proposition forthe AK structure (J,Ω,g) such that π is J-holomorphic.

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Classification of T 2-bundles over T2

By Sakamoto and Fukuhara the diffeomorphic classes ofT 2-bundles over T2 are classified in 8 families:

G Structure equations of Xi), ii) SO(4) nR4 (0,0,0,0)

iii), iv), v) Nil3 × S1 (0,0,0,12)vi) Nil4 (0,13,0,12)

vii), viii) Sol3 × R (0,0,13,41)

The Lie group G is the geometry type of Γ\X .• In the cases different from iii) the fibration of M4 as torusbundle is unique.• In the case iii) one has two fibrations

πxy : M4 −→ T2xy , πyt : M4 −→ T2yt .

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Theorem (Geiges)

Let M4 = Γ\X be an orientable T 2-bundle over a T2. Then• M4 has a symplectic form and every class a ∈ H2(M4,R)

with a2 6= 0 can be represented by a symplectic form;• M4 has a Kähler structure if and only if X = R4;• If X = Nil4 then every invariant AK structure on M4 is

Lagrangian;• If X = Sol3 ×R, then every invariant AK structure on M4 is

not Lagrangian.

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References

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Classification of the invariant AK structures

Goal: Classify invariant (Lagrangian) AK structures on Nil3 × Rand Nil4.

Both Nil3 × R and Nil4 have an invariant coframe (ei ) such that

de1 = 0 , de2 = �e1 ∧ e3 , de3 = 0 , de4 = e1 ∧ e2 .

Lemma

Let (g, J,Ω) be an invariant AK structure on Nil3 ×R or on Nil4.Then there exists an orthonormal basis (f i ) for which

Ω = f 14 + f 23,

and f 1 ∈〈e1〉, f 2 ∈

〈e1,e2

〉, f 3 ∈

〈e1,e2,e3

〉.

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• The case X = Nil3 × R


iii), iv), v) Nil3 × S1 (0,0,0,12)vi) Nil4 (0,13,0,12)

vii), viii) Sol3 × R (0,0,13,41)

• The total space M4 could be also a infra-nilmanifold.• The invariant AK structures on M4 could be either Lagrangianor non-Lagrangian and the argument used in the KT case hasto be modified.

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For these other Lagrangian cases the CY equation reduces toa generalized Monge Ampere equation of the type

(a + pxx − l11px −m11py )(b + pyy − l22px −m22py )−(c + pxy − l12px −m12py )2 = eF ,

for p ∈ C∞(T2) satisfying(a + pxx − l11px −m11py c + pxy − l12px −m12pyc + pxy − l12px −m12py b + pyy − l22px −m22py

)> 0,

with (a cc b

)> 0, m11l22 = 0, l11l22 − l212 = 0,

m11m22 −m212 = 0, l11m22 + l22m11 − 2l12m12 = 0.

Calabi-Yau equation

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References

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• The case X = Nil4


iii), iv), v) Nil3 × R (0,0,0,12)vi) Nil4 (0,13,0,12)

vii), viii) Sol3 × R (0,0,13,41)

In this case the total spaces are nilmanifolds, all the invariantAK structure are Lagrangian and the CY equation reduces to ageneralized M-A equation of the previous type.

Calabi-Yau equation

Anna Fino










References

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Solutions of the generalized MA equation

Goal: Show that

(a + pxx − l11px −m11py )(b + pyy − l22px −m22py )−(c + pxy − l12px −m12py )2 = eF ,

with ∫T2

(eF − ab + c2) = 0

has a solution on T2.

• The first step consists to show that the solutions of thegeneralized M-A equation are unique modulo addition of aconstant.

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References

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• To prove the existence we apply the continuity method to theequation (∗)t on T2

(a + pxx − tl11px − tm11py )(b + pyy − tl22px − tm22py )−

(c + pxy − tl12px − tm12py )2 = teF + (1− t)(ab − c2),

t ∈ [0,1], together with the condition (∗∗)t(a + pxx − tl11px − tm11py c + pxy − tl12px − tm12pyc + pxy − tl12px − tm12py b + pyy − tl22px − tm22py

)> 0

which requires a priori extimates first.

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• The equation (∗)t is elliptic, all date are C∞, then its C2solutions are smooth.

• t = 0 −→ (a + pxx )(b + pyy )− (c + p2xy ) = ab − c2:solution p ≡ 0.

• t = 1 −→ the equation that we want to solve.• Let I = {t ∈ [0,1] | (∗t ) has solution p satisfying (∗∗t )}.

• 0 ∈ I and I 6= ∅.

• We show that I is open and closed in [0,1].

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• I is open and closed.

Let 0 < µ < 1 and X 2,µ := {p ∈ C2,µ(T2) |∫T2 p = 0}

Consider St : X 2,µ → X 0,µ

St (p) := (a + pxx − tl11px − tm11py )(b + pyy − tl22px − tm22py )−(c + pxy − tl12px − tm12py )2 −

[teF + (1− t)(ab − c2)

].

If p ∈ X 2,µ satisfies (∗∗)t then

dSt [p] : X 2,µ → X 0,µis an isomorphism.

This implies that I is open.

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An a priori estimate: Let p ∈ C2(T2) a solution of (∗)t and (∗∗)t .Then, p ∈ C∞(T2) and for any positive integer k

‖p −∫T2

p‖Ck (T2) ≤ C

where C = C(k ,F ,ab, c, (lij ), (mij )).

In particular if∫T2 p = 0 =⇒ ‖p‖Ck (T2) ≤ cost

Let I 3 tn −→ t0, then we have a sequence ptn ∈ X 2,µ such thatStn (ptn ) = 0

Since ‖ptn‖Ck (T2) is bounded, then there is a subsequence ptlwhich converges to pt0 ∈ S

−1t0 (0).

Since St is elliptic for every t , pt0 is smooth.

Hence I is closed!

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References

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References

S.K.Donaldson, in Inspired by S.S.Chern, World Sci. 2006.A. Fino, Y.Y. Li, S. Salamon, L. Vezzoni, to appear in Trans.AMSH. Geiges, Duke Math. J., 1992.K.Kodaira, Amer. J. Math., 1964.Y.Y. Li, Comm. Pure Appl. Math., 1990.K. Sakamoto, S. Fukuhara, Tokyo J. Math., 1983.V. Tosatti, B. Weinkove, S.T. Yau, Proc. London Math. Soc.,2008.V. Tosatti, B. Weinkove, J. Inst. Math. Jussieu , 2011.

BackgroundSymplectic CY problemUniqueness of solutionsExistence of solutions

The CY equation on the KT manifoldAK structure on the KT manifoldTosatti and Weinkove resultCY equation on the KT manifold II

Main resultClassification of T2-bundles over T2Classification of the invariant AK structuresSolutions of the generalized MA equation

References

calabi-yau equation anna fino calabi-yau equation on ...the kt manifold ak structure on the kt...

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