Geometria Simplettica e metriche Hermitiane...

Taming symplectic forms and SKT geometryThe Symplectic Calabi-Yau problem on T 2-bundles over T2

References

Geometria Simplettica e metriche Hermitianespeciali

Anna Fino

Dipartimento di MatematicaUniversita di Torino

1 Marzo 2013

Anna Fino Geometria Simplettica e metriche Hermitiane speciali


References

1 Taming symplectic forms and SKT geometryLink with SKT metricsThe pluriclosed flowResults on Nilmanifolds

2 The Symplectic Calabi-Yau problem on T 2-bundles over T2

Kodaira-Thurston manifoldMain resultThe Monge-Ampere equation

3 References



References

Link with SKT metricsThe pluriclosed flowResults on Nilmanifolds

Tamed almost complex structures

Definition

An almost cx structure J on a symplectic manifold (M2n,Ω) istamed by Ω if Ω(X , JX ) > 0, ∀X 6= 0.

If J is tamed by Ω, then

g(X ,Y ) =1

2(Ω(X , JY )− Ω(JX ,Y ))

is a J-Hermitian metric.



References


Theorem (Streets, Tian; Li, Zhang)

If a compact complex (M4, J) admits a symplectic structuretaming J, then (M4, J) has a Kahler metric.

Problem

Does there it exist an example of a compact complex (M2n, J),with n > 2, admitting a symplectic form taming J, but no Kahlerstructures?

We will give some negative answer to this problem.



References


Link with SKT metrics

• Giving a symplectic form Ω taming a complex structure J on M2n

is equivalent to assign a J-Hermitian g whose fundamental 2-formF = g(J·, ·) satisfies ∂F = ∂β, for some ∂-closed (2, 0)-form β.

Thus in particular ∂∂F = 0.

Definition

A J-Hermitian metric g on a complex manifold (M2n, J) is said tobe strong Kahler with torsion (SKT) or pluriclosed if ∂∂F = 0.



References


Bismut connection

Proposition (Bismut)

On any Hermitian (M2n, J, g) ∃! connection ∇B such that

∇Bg = 0 (metric)∇BJ = 0 (Hermitian)c(X ,Y ,Z ) = g(X ,TB(Y ,Z )) 3-form

where TB is the torsion of ∇B .

∇B = ∇LC + 12 c is the Bismut connection and c = −JdJF .

c = 0 ⇐⇒ ∇B = ∇LC ⇐⇒ (M2n, J, g) is Kahler

(J, g) on M2n is SKT if and only if dc = 0.



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The pluriclosed flow

Let (M2n, J, g0) be an Hermitian manifold. Streets and Tianintroduced the pluriclosed flow

∂F (t)∂t = Φ(F (t)),

F (0) = F0,

where Φ(F ) = −∂∂∗F − ∂∂∗F − i2∂∂ log det g = −(ρB)1,1(F ).

Proposition (Streets,Tian)

Let (M2n, J, g) be a SKT manifold. Then F → Φ(F ) is a realquasi-linear second-order elliptic operator when restricted toJ − Hermitian SKT metrics.

If g(0) is SKT (Kahler), then g(t) is SKT (Kahler) for all t.



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Definition (Streets, Tian)

A SKT metric g on compact (M2n, J) is static if Φ(F ) = λF , orequivalently if (ρB)1,1 = λF , for a constant λ.

If g is Kahler and static, then it is Kahler-Einstein.

Proposition (Streets,Tian)

Let (M2n, J) be compact with a static SKT metric g . If λ 6= 0,then F = Ω1,1 with Ω is a symplectic form Ω taming J.



References


Given a symplectic form Ω0 taming a complex structure J on M2n

we can define the HS flow∂Ω(t)∂t = −ρB(Ω1,1(t))

Ω(0) = Ω0.

• We have a short-time existence of solutions and Ω(t) is taming Jfor every t.

Problem

Study the properties of previous HS flow.

Still in progress!



References


Results on nilmanifolds

Theorem (Enrietti, -, Vezzoni)

(G/Γ, J) with J left-invariant and G any Lie groupIf Jξ ∩ [g, g] 6= 0, then (G/Γ, J) does not admit any compatibleHermitian-symplectic structure.


M2n = G/Γ nilmanifold (not a torus), J left-invariant.1) If (M2n, J) has a SKT metric, then G has to be 2-step and(M2n, J) is a principal holomorhic torus bundle over a torus.2) (M2n, J) does not admit any symplectic form taming J.

Remark

2) was proved for n = 3 by Angella and Tomassini.



References


To prove 1) we show that J has to preserve the center ξ of g andthat a SKT structure on g induces a SKT structure on g/ξ.

To prove 2) we may assume that the Hermitian-symplectic form isinvariant using the symmetrization process.

Since J preserves the center ξ and [g, g] ⊆ ξ, we have thatJξ ∩ [g, g] 6= 0 and so (g, J) does not admit any symplectic formtaming J.



References


The Hermitian flow on nilmanifolds


Let (G/Γ, J) be a complex nilmanifold with an invariant SKTstructure F0. Then the pluriclosed flow

∂F (t)∂t = −(ρB)1,1(F (t)),

F (0) = F0,

has a unique solution F (t) defined for every t ∈ R+.

To prove the theorem we use that the pluriclosed flow on a SKTLie algebra can be studied using the Lauret trick and deformingthe Lie bracket instead of the Hermitian metric.



References


Let (R2n, J0, < , >) and

N2 = µ ∈ Λ2 ⊗ Λ1 | µ(µ(·, ·), ·) = 0,Nµ = 0.∂F (t)∂t = (dd∗F )1,1(t),

F (0) = F0,⇐⇒

∂µ(t)∂t = 1

2δµPµ(t),

µ(0) = µ0,

where

δµPµ = µ(Pµ·, ·)+µ(·,Pµ·)−Pµµ(·, ·), < Pµ(Zi ),Zj >= ρB(Zi ,Zj).

We have

< δµPµ, µ >= −8 < Pµ,Pµ >≤ 0⇒ ∂∂t < µ(t), µ(t) >≤ 0.



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Calibrated almost complex structures

Definition

An almost cx structure J on a symplectic manifold (M2n,Ω) iscalibrated by Ω (or Ω is compatible with J) if J is tamed andΩ(JX , JY ) = Ω(X ,Y ), ∀X ,Y .

• If J is calibrated by Ω =⇒ (Ω, J) is an almost-Kahler (AK)structure ⇒ g(X ,Y ) = Ω(X , JY ) is a J-Hermitian metric.

• If J is integrable, then the AK structure (Ω, J, g) is Kahler.



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The Symplectic Calabi-Yau problem

Theorem (Yau, Symplectic version)

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

∫M2n Ωn =

∫M2n σ. Then there exists a

unique Kahler form Ω with [Ω] = [Ω] such that

Ωn = σ ←→ CY Equation

CY equation ↔

(Ω + dα)n = ef Ωn,J(dα) = dα

↔ (Ω + ddch)n = ef Ωn.



References


(Ω + ddch)n = ef Ωn is a Monge-Ampere equation in h.

CY equation ↔

(Ω + ddch)n = ef Ωn,∫M2n h Ωn = 0.

(∗)

Yau’s theorem: (∗) has always a unique solution h.

Donaldson introduced the symplectic version of the CY equationfor AK manifolds.



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Symplectic CY problem

Let (M2n, J,Ω, g) be a compact AK manifold with a volume formσ = ef Ωn satisfying

∫M2n e

f Ωn =∫M2n Ωn. Then

CY equation←→

(Ω + dα)n = ef Ωn

J(dα) = dα(∗)

(∗) is elliptic for n = 2 and the solutions are unique[Donaldson];

(∗) is overdetermined for n > 2.



References


Problem

Can the Yau Theorem be generalized to AK 4-manifolds?

On a AK (M4,Ω, J, g) ∃! connection ∇C (the canonical or Chernconnection) such that ∇CJ = ∇CΩ = 0 , Tor1,1(∇C ) = 0 .

Theorem (Tosatti,Weinkove,Yau)

If R(g , J) ≥ 0, then the Calabi-Yau problem can be solved forevery normalized volume form on (M4,Ω, J, g), where R(g , J) is

defined by Ri jkl = R j

ikl+ 4N r

l jN irk.

• The theorem can be applied to an infinitesimal deformation ofthe Fubini-Study Kahler structure on CPn but it cannot be appliedfor instance to the Kodaira-Thurston surface!

• We will study the CY problem on T 2-bundles over T2.



References


The CY equation on the Kodaira-Thurston manifold

Remark

The existence result by Tosatti - Weinkove-Yau cannot be appliedto the KT manifold M4 = (Γ\Nil3)× S1.

Nil3 =

1 x z

0 1 y0 0 1

, x , y , z ∈ R

• M4 has a global invariant coframe e i such that

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

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



References


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

T 2 = S1×S1 → Γ\Nil3 ×S1

↓ πxyT2xy

• M4 has the Lagrangian (with respect to πxy ) AK structure

Ω = e1 ∧ e4 + e2 ∧ e3, g =4∑

i=1

e i ⊗ e i ,

i.e. Ω vanishes on the fibers.



References


Theorem (Tosatti,Weinkove)

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

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

Then dα = vxx e23 + vxy (e13 + e24) + vyy e14 and the CY equation(Ω + dα)2 = ef Ω2 becomes the Monge-Ampere equation

(1 + vxx)(1 + vyy )− v 2xy = ef

Theorem (Li)

The Monge-Ampere equation on the standard torus T n has alwaysa solution.



References


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.



References


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

Theorem (Ue)

Every orientable T 2-bundle over a 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 covered bya solvmanifold or equivalently a quotient Γ\X , where the discretegroup Γ contains a lattice Γ of X such that Γ\Γ is finite.



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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 (M4 = Γ\X , J,Ω, g) withan invariant AK structure, the Tosatti-Weinkove-Yau conditionR(g , J) ≥ 0 is satisfied if and only if (Ω, J) is Kahler.



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The main result

Theorem (–, Li, Salamon, Vezzoni / Buzano, –, Vezzoni)

Let M4 = Γ\X be a T 2-bundle over a T2 endowed with aninvariant AK structure (J,Ω, g). Then for every normalizedT 2-invariant volume form σ = eF Ω2, F ∈ C∞(T2) the associatedCY problem has a unique solution.

Layout of the proof:

Use the classification of T 2-bundles of T2;

Classify in each case invariant Lagrangian AK structures andinvariant non-Lagrangian AK structures;

Rewrite the problem in terms of a Monge-Ampere equation;

Show that such an equation has a solution.



References


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 X

i), 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 −→ T2

yt .



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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) witha2 6= 0 can be represented by a symplectic form;

M4 has a Kahler structure if and only if X = R4;

If X = Nil4 then every invariant AK structure on M4 isLagrangian;

If X = Sol3 × R every invariant AK structure on M4 isnon-Lagrangian.



References


Classification of invariant AK structures

Goal: Classify all invariant AK structures (g ,Ω) on Nil3 × R, Nil4,Sol3 × R.

In each case there exists an ON basis (f i ) such that Ω = f 12 + f 34

and

• G = Nil4 → f 1 ∈< e1 >, f 2 ∈< e1, e2 >, f 3 ∈< e1, e2, e3 > .

• G = Sol3 × R → f 1 ∈< e1 >, f 3 ∈< e3 >, f 4 ∈< e3, e4 > .

• G = Nil4 → f 1 ∈< e1 >, g(e3, f 2) = 0, g(e3, f 3)g(e4, f 4) ≥ 0.



References


• The case X = Nil3 × R




• In this case all the total spaces M4 are nilmanifolds.

• All the invariant AK structures are Lagrangian and we can workas for the Kodaira-Thurston surface.



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• The case X = Nil3 × R




• The total space M4 could be also a infra-nilmanifold.

• The invariant AK structures on M4 could be either Lagrangian ornon-Lagrangian and the argument used for the Kodaira-Thurstonsurface has to be modified.



References


• The case X = Sol3 × R


i), ii) SO(4) nR4 (0, 0, 0, 0)iii), iv), v) Nil3 × R (0, 0, 0, 12)


In this case the total space could be an infra-sovmanifold, allinvariant AK structures are non-Lagrangian and the CY equationreduces to a Monge-Ampere equation.



References


• The case X = Nil4


i), ii) SO(4) nR4 (0, 0, 0, 0)iii), iv), v) Nil3 × R (0, 0, 0, 12)


In this case the total spaces are nilmanifolds, all the invariant AKstructure are Lagrangian and the CY equation reduces to the sameMonge-Ampere equation for Lagrangian AK structures in thefamilies iv) and v) associated to Nil3 × R.



References


The Monge-Ampere equation

The following equation covers all cases

A11[u]A22[u]− (A12[u])2 = E1 + E2ef ,

whereA11[u] = uxx + B11ux + C11 + Du,A12[u] = uxy + B12uy + C12,A22[u] = uyy + B22uy + C22

and Bij ,Cij ,D,Ei are constants.

In the Lagrangian case D = 0.



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Solutions to the Monge-Ampere equation

Goal: Show that A11[u]A22[u]− (A12[u])2 = E1 + E2ef has asolution on T2.

• The first step consists to show that the solutions to the equationare unique up to a constant.

• We look for a solution u satisfying∫T2 u = 0.

• We apply the continuity method to

A11[u]A22[u]− (A12[u])2 = E1 + (1− t)E2 + tE2ef , t ∈ [0, 1].

using a priori estimate

‖u‖C2 ≤ 2(B11 + 1)|B22|e2C22 + C11 + C22.



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Open related problems

• Find a (generalized) ∂∂-lemma which ensures that thesymplectic CY problem reduces to a Monge-Ampere equation.

• Find a proof of the main theorem in terms of a (modified) Ricciflow.

• Find examples of compact AK (non Kahler) manifolds withR(g , J) > 0.

• Introduce and study a CY equation for other geometric structures(positive 3-forms in dimension 6, G2-structures with torsion indimension 7, Spin(7)-structures with torsion in dimension 8...).



References

References

E. Buzano, A. Fino, L. Vezzoni, The Calabi-Yau equation forT 2-bundles over T 2: The non-Lagrangian case, Rend. Semin.Mat. Univ. Politec. Torino (2011).

N. Enrietti, A. Fino, L. Vezzoni, Tamed symplectic forms andstrong Kahler with torsion metrics, J. Sympl. Geom. (2012).

N. Enrietti, A. Fino, L. Vezzoni, The pluriclosed flow onnilmanifolds and Tamed symplectic forms, arXiv:1210.4816

A. Fino, Y.Y. Li, S. Salamon, L. Vezzoni, The Calabi-Yau equationon 4-manifolds over 2-Tori , Trans. Amer. Math. Soc. (2013).



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