Examples and counter-examples of log-symplectic manifolds › ... › madrid2014.pdf · Examples and counter-examples of log-symplectic manifolds Gil Cavalcanti Utrecht University

Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing

Examples and counter-examples oflog-symplectic manifolds

Gil Cavalcanti

Utrecht University

November 2014Madrid

Log-symplectic Cavalcanti


Poisson geometry

A Poisson structure is π ∈ Γ(∧2TM) such that

[π, π] = 0.



Poisson geometry

Example (Symplectic manifolds)

If ω ∈ Ω2(M) is symplectic

ω : TM∼=−→ T∗M ⇔ π := ω−1 : T∗M

∼=−→ TM.

dω = 0 ⇔ [π, π] = 0.

Example (Trivial Poisson structure)On any manifold, take π = 0.



Poisson geometry

ProblemFind a large class of Poisson manifolds which share importantfeatures of symplectic manifolds.



Poisson geometry

DefinitionA log-symplectic structure on M2n is a Poisson structure π suchthat πn only has nondegenerate zeros.

AimFind many examples of log-symplectic manifolds and topologicalobstructions to their existence.



Outline of Topics

1 Poisson geometry

2 Local forms and invariants

3 Topology

4 Constructing by deformations

5 Surgery

6 Achiral Lefschetz Fibrations

7 Reversing the surgery



Poisson structures are also Dirac structures

TM⊕ T∗M is endowed withthe natural pairing

〈X + ξ,Y + η〉 =12

(η(X) + ξ(Y))

the Courant bracket on Γ(TM⊕ T∗M):

[[X + ξ,Y + η]] = [X,Y] + LXη − ιYdξ.




(Definition 2) A Poisson structure is a maximal isotropicinvolutive subbundle L ⊂ TM⊕ T∗M such that L ∩ TM = 0:

L = π(ξ) + ξ : ξ ∈ T∗M.




(Definition 3) A Poisson structure is a line K ⊂ ∧•T∗Mgenerated (locally) by a form ρ ∈ Γ(K) pointwise of the form

eω ∧ θ,

such that1 θ is a decomposable form;2 ω ∈ Ω2(M);

3 (eω ∧ θ)top = ωn−k

2 ∧ θ 6= 0;4 There is X ∈ X(M) such that

dρ = ιXρ.



Log-symplectic structures

K∗ has a natural section

s : K −→ R×M; s(ρ) = ρ0 ∈ Ω0(M).

s has nondegenerate zeros⇔ πn has nondegenerate zeros.

DefinitionA log-symplectic structure on M2n is a Poisson structure suchthat s only has nondegenerate zeros.Z = [s = 0] is the singular locus of π.M\Z is the symplectic locus of π.A bona fide log-symplectic structure is one with Z 6= ∅



Log-symplectic structures

ExampleEvery surface admits a log-symplectic structure.



Local form 1

Lemma (Guillemin–Miranda–Pires)Let p ∈ Z, then in a neighbourhood U of p, π is diffeomorphic to thePoisson structure

x1∂x1 ∧ ∂y1 + ∂x2 ∧ ∂y2 + · · ·+ ∂xn ∧ ∂yn ,

i.e., π is the product of a log-symplectic structure in R2 with asymplectic structure σ ∈ Ω2(R2n−2).

In U, π is equivalent to the line bundle

eσ(x + dx ∧ dy) = eσ+d log |x|∧dy.



Local form 2Theorem (Guillemin–Miranda–Pires)

Let (M2n, π) be a log-symplectic manifold and Z be the zero locus of s.Then

1 N Z is isomorphic to KM|Z ∼= ∧2nT∗M|Z,2 the Poisson structure on M induces a Poisson structure on Z

and singles out a nowhere vanishing closed section of KZ

ρZ = eσ ∧ θ,

where θ is a degree 1 form, σ is closed and well defined up to aterm of the form df ∧ θ.

Conversely, if two log-symplectic structures have diffeomorphiccompact singular loci, Z, and induce the same data then they arediffeomorphic in a neighbourhood of Z.



Log-symplectic geometry — local forms

Theorem (Rephrased)Let (M, π) be log-symplectic, Z a compact component of the singularlocus. Then, in a neighbourhood of Z, π determines and is determinedby:

1 N Z as a vector bundle, i.e., w1 ∈ H1(Z,Z2).2 A closed 1-form θ ∈ Ω1(Z).3 A closed 2-form σ ∈ Ω2(Z) defined up to the addition of df ∧ θ

such thatθ ∧ σn−1 6= 0.




DefinitionA cosymplectic structure on Z2n−1 is a pair of closed formsθ ∈ Ω1(Z) and σ ∈ Ω2(Z) such that

θ ∧ σn−1 6= 0.




Corollary (Guillemin–Miranda–Pires (2012))Any log-symplectic structure inducing the cosymplectic structure(σ, θ) on Z is equivalent to a neighbourhood of the zero section of N Zendowed with the structure

d log |x| ∧ θ + σ,

where |x| is the distance to the zero section measured with respect to afixed fiberwise linear metric on N Z.




DefinitionA component, Z, of the singular locus is proper if it is compactand has a compact symplectic leaf.

pφ(p)

F

X

Z = R× F/Z (t, p) ∼ (t + λ, ϕ(p)),

for a period λ ∈ R and a symplectomorphism ϕ : F −→ F.




Theorem (Marcut–Osorno-Torres (2013))A log-symplectic structure with compact singular locus Z can bedeformed into a proper one.



Topology

Theorem (Marcut–Osorno-Torres (2013))

If the singular locus of a log-symplectic manifold M2n has a compactcomponent, then there is a ∈ H2(M) such that

an−1 6= 0.



Topology

Proof.Let ψ : [0, δ] −→ [0, 1] be a bump function. For each componentof the singular locus modify the log-symplectic form to

d(ψ(|x|) log |x|) ∧ θ + σ

This gives a globally defined closed 2-form ω and

θ ∧ ωn−1|Z = θ ∧ σn−1 6= 0 ∈ H2n−1(Z).

[ω|n−1Z ] 6= 0.

Let a = [ω].



Topology

Corollary (Marcut–Osorno-Torres (2013))Nonabelian compact connected Lie groups have no log-symplecticstructure.

Corollary (Marcut–Osorno-Torres (2013))If n1 > 2, Sn1 × · · · × Snk has no log-symplectic structure.



Topology

Theorem (Cavalcanti (2013))

If M2n is a compact oriented bona fide log-symplectic manifold, thenthere is b ∈ H2(M) such that

b2 = 0 and b ∪ an−1 6= 0.

Proof.Deform the structure into a proper one. Then each fiber F ofZ −→ S1 is homologically nontrivial in M since

〈an−1, [F]〉 6= 0,

hence b = PD(F) 6= 0. Within Z, F has no self intersection sob2 = 0. Finally, 〈b ∪ an−1, [M]〉 = 〈an−1, [F]〉 6= 0.



Topology

Corollary (Cavalcanti (2013))

In a compact orientable bona fide log-symplectic manifold M2n,b2i(M) ≥ 2 for 0 < i < n.

Corollary (Cavalcanti (2013))For n > 1, CPn has no bona fide log-symplectic structure.

Corollary (Cavalcanti (2013))The blow-up of CPn along a symplectic submanifold of codimensiongreater than 4 has no bona fide log-symplectic structure (McDuffexamples).



Topology

Corollary (Cavalcanti (2013))A compact, orientable four manifold with definite intersection formhas no bona fide log-symplectic structure.

Corollary (Cavalcanti (2013))#kCPn has no log-symplectic structure if n > 1.



Deformations


Let (M2n, ω) be a symplectic manifold and F2n−2 ⊂M a compactsymplectic submanifold with trivial normal bundle. Then M has logsymplectic structures with an arbitrary number of singular loci.



Deformations

Proof.Symp nhood thm⇒N F = D2 × F with product structure.Deform the Poisson structure on D2

πD = ∂r2 ∧ ∂θ ; πD = f (r)∂r2 ∧ ∂θ,

where the graph of f is

1

0 ε

...

Set ω = π−1D + ωF.



Deformations


The blow-up of CP2 at a point has a bona fide log-symplecticstructure.


For m,n > 0, #mCP2#nCP2 has a log-symplectic structure.



Deformations

#mCP2#nCP2 symplectic bona fide log-symplectic

m > 1, n > 0 % X

m > 1, n = 0 % %

m = 1, n > 0 X X

m = 1, n = 0 X %

m = 0, n > 0 % %



Surgery

Building block: Let (Z, σ, θ) be a co-symplectic manifold, let

N = (−1, 1)× Z; ωN = d log |x| ∧ θ + σ.

Z embeds in the symplectic locus of N as −ε × Z andωN |Z = σ.Given a function f : Z −→ R we have a second disjointembedding of Z in the symplectic locus of N :

p ∈ Z 7→ (eε′f (p), p).

For this embedding ω|Z = σ + df ∧ θ.



SurgeryBuilding block

Z (singular locus)

Z(coisotropic

submanifold)

Z(coisotropic

submanifold)

-1 10



Surgery

Ingredients: A log-symplectic manifold (M2n, π) together withtwo embeddings of a compact, connected, cosymplecticmanifold (Z, σ, θ), ιi : Z2n−1 →M, such that

1 Each ιi(Z) is a separating submanifold of M lying in thesymplectic locus of π and ι1(Z) ∩ ι2(Z) = ∅;

2 There is f ∈ C∞(Z) such that ι∗1ω = ι∗2ω − df ∧ θ = σ, whereω is the symplectic form on the symplectic locus of M.



SurgeryIngredient

(Z,σ)(coisotropicsubmanifold)



SurgeryIngredient:

(Z,σ)(coisotropicsubmanifold)

Interior

Exterior



Surgery

The Surgery:

(Z,σ)(coisotropic

submanifold)Z

(singular locus)Z

(coisotropicsubmanifold)

Z(coisotropic

submanifold)

Interior

Interior Interior



SurgeryThe Surgery:

(Z,σ)(coisotropic

submanifold)Z

(singular locus)Z

(coisotropicsubmanifold)

Interior

Interior



SurgeryIngredient 2

(Z,σ + dfθ)(coisotropicsubmanifold)

Interior Exterior



SurgeryIngredient 2


Interior



SurgeryThe surgery

(Z,σ)(coisotropic

submanifold)Z

(singular locus)

Interior


Interior



Surgery

Theorem (Cavalcanti 2013)The manifold

M = MI1 ∪Z MI

2,

has a log-symplectic structure whose singular locus contains theidentified boundaries.



Example: n#S2 × S2

Let E(n) −→ CP1 be the elliptic surface with 12n nodal fibers.E(1) = CP2#9 ¯CP2;E(2) = K3.In p : E(2n) −→ CP1, take an embedded circle γ : S1 −→ CP1

and let ΣI be the interior region of CP1 determined by γ

a

b

a

a

b

b

a

b

...

...

γ



Example: n#S2 × S2

Then Z = p−1(γ(S1)) is a proper cosymplectic manifold.

p−1(ΣI) ∪∂ p−1(ΣI)

has a log-symplectic structure.Four-manifold theory tells us that this is n#S2 × S2 where n + 1is the number of singular values in ΣI.

Corollary

For all n > 0, n#S2 × S2 has a log-symplectic structure.



Achiral Lefschetz Fibrations

DefinitionAn achiral Lefschetz fibration is a smooth map p : M2n −→ Σ2

between compact manifolds such thatthe pre-image of any critical value has only one criticalpoint andfor any such pair of critical value and critical point thereare complex coordinates which render

p(z1, · · · , zn) = z21 + · · ·+ z2

n.





Let M4 and Σ2 be compact manifolds and p : M −→ Σ be an achiralLefschetz fibration for which the fibers are orientable and represent anontrivial real homology class. Then M has a bona fide log-symplecticstructure.

Proof.Adapt Gompf’s construction of symplectic four manifolds fromLefschetz fibrations.




Remarks‘The fibers are orientable and represent a nontrivial real

homology class’

1 If p : M −→ Σ is an achiral Lefschetz fibration and M and Σare orientable, then the fibers are orientable.

2 If the generic fiber has genus different from 1, it representsa nontrivial real homology class.

3 If Σ is orientable and the fibration has a section, then thefiber represents a nontrivial homology class.

4 S4 admits an achiral Lefschetz fibration over S2 whosegeneric fibers are tori.




Theorem (Etnyre–Fuller (2006))

Let M4 be a compact oriented 4-manifold. Then there is a framedcircle in M such that the manifold obtained by surgery along thatcircle admits an achiral Lefschetz fibration with base CP1. Moreoverthis fibration admit sections.

Theorem (Etnyre–Fuller (2006))

Let M4 be compact and simply connected. Then M4#CP2#CP2 andM4#(S2 × S2) admit an achiral Leftschetz fibration with a section.





Let M4 be a compact oriented 4-manifold. Then there is a framedcircle in M such that the manifold obtained by surgery along thatcircle admits a bona fide log-symplectic structure.


Let M4 be compact and simply connected. Then M4#CP2#CP2 andM4#(S2 × S2) admits bona fide log-symplectic structures.



Reversing the Surgery

QuestionDoes every compact compact oriented log-symplectic manifold arisefrom the surgery?

The answer is yes if every co-symplectic manifold appears as aseparating submanifold of a compact symplectic manifold.



Reversing the Surgery

In four dimensions this is the case:

Theorem (Kronheimer–Mrowka)Every compact co-symplectic three-manifold Z appears as aseparating submanifold of a compact symplectic manifold M4.


Let (M4, π) be a compact, orientable, log-symplectic manifold. Theneach unoriented component of M\ Z can be compactified as asymplectic manifold.


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Examples and counter-examples of log-symplectic manifolds › ... › madrid2014.pdf · Examples and counter-examples of log-symplectic manifolds Gil Cavalcanti Utrecht University