Post on 28-Jun-2020
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 Local forms Topology Deformations Surgery Fibrations Reversing
Poisson geometry
A Poisson structure is π ∈ Γ(∧2TM) such that
[π, π] = 0.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Poisson geometry
ProblemFind a large class of Poisson manifolds which share importantfeatures of symplectic manifolds.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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
Log-symplectic Cavalcanti
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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.
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Poisson structures are also Dirac structures
(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.
Log-symplectic Cavalcanti
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Poisson structures are also Dirac structures
(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 Cavalcanti
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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 Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Log-symplectic structures
ExampleEvery surface admits a log-symplectic structure.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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 Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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.
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Log-symplectic geometry — local forms
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Log-symplectic geometry — local forms
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Log-symplectic geometry — local forms
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Log-symplectic geometry — local forms
Theorem (Marcut–Osorno-Torres (2013))A log-symplectic structure with compact singular locus Z can bedeformed into a proper one.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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.
Log-symplectic Cavalcanti
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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].
Log-symplectic Cavalcanti
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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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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).
Log-symplectic Cavalcanti
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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.
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Deformations
Theorem (Cavalcanti (2013))
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.
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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.
Log-symplectic Cavalcanti
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Deformations
Corollary (Cavalcanti (2013))
The blow-up of CP2 at a point has a bona fide log-symplecticstructure.
Corollary (Cavalcanti (2013))
For m,n > 0, #mCP2#nCP2 has a log-symplectic structure.
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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 % %
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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.
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SurgeryBuilding block
Z (singular locus)
Z(coisotropic
submanifold)
Z(coisotropic
submanifold)
-1 10
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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.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
SurgeryIngredient
(Z,σ)(coisotropicsubmanifold)
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
SurgeryIngredient:
(Z,σ)(coisotropicsubmanifold)
Interior
Exterior
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Surgery
The Surgery:
(Z,σ)(coisotropic
submanifold)Z
(singular locus)Z
(coisotropicsubmanifold)
Z(coisotropic
submanifold)
Interior
Interior Interior
Log-symplectic Cavalcanti
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SurgeryThe Surgery:
(Z,σ)(coisotropic
submanifold)Z
(singular locus)Z
(coisotropicsubmanifold)
Interior
Interior
Log-symplectic Cavalcanti
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SurgeryIngredient 2
(Z,σ + dfθ)(coisotropicsubmanifold)
Interior Exterior
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
SurgeryIngredient 2
(Z,σ + dfθ)(coisotropicsubmanifold)
Interior
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
SurgeryThe surgery
(Z,σ)(coisotropic
submanifold)Z
(singular locus)
Interior
(Z,σ + dfθ)(coisotropicsubmanifold)
Interior
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
Surgery
Theorem (Cavalcanti 2013)The manifold
M = MI1 ∪Z MI
2,
has a log-symplectic structure whose singular locus contains theidentified boundaries.
Log-symplectic Cavalcanti
Poisson geometry Local forms Topology Deformations Surgery Fibrations Reversing
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
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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.
Log-symplectic Cavalcanti
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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.
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Achiral Lefschetz Fibrations
Theorem (Cavalcanti (2013))
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.
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Achiral Lefschetz 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.
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Achiral Lefschetz Fibrations
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.
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Achiral Lefschetz Fibrations
Corollary (Cavalcanti (2013))
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.
Corollary (Cavalcanti (2013))
Let M4 be compact and simply connected. Then M4#CP2#CP2 andM4#(S2 × S2) admits bona fide log-symplectic structures.
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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.
Log-symplectic Cavalcanti
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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.
Theorem (Cavalcanti (2013))
Let (M4, π) be a compact, orientable, log-symplectic manifold. Theneach unoriented component of M\ Z can be compactified as asymplectic manifold.
Log-symplectic Cavalcanti