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TRANSCRIPT
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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
1
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
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
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
-
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
3
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.
-
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
3
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.
-
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
4
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
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
5
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
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
6
Uniqueness of solutions
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 .
-
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
6
Uniqueness of solutions
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 .
-
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
7
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.
-
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
8
Existence of solutions
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.
-
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
8
Existence of solutions
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.
-
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
9
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).
-
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
10
• 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).
-
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
11
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.
-
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
12
Tosatti and Weinkove result
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
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
12
Tosatti and Weinkove result
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
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
13
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.
-
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
14
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.
-
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
14
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.
-
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
15
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
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
15
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
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
16
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
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
16
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
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
17
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.
-
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
17
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.
-
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
18
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 .
-
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
19
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.
-
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
20
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
〉.
-
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
21
• The case X = Nil3 × R
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 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.
-
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
22
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
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
23
• The case X = Nil4
G Structure equations of Xi), ii) SO(4) nR4 (0,0,0,0)
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
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
24
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.
-
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
25
• 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.
-
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
26
• 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].
-
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
27
• 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.
-
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
28
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!
-
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
29
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