Rui Loja Fernandes (based on joint work with Ioan Marcut˘2020/04/09 · Local models around...

Local models around Poisson submanifolds

Rui Loja Fernandes...

(based on joint work with Ioan Marcut,)

Department of MathematicsUniversity of Illinois at Urbana-Champaign, USA

April 9, 2020

Overview - Main ProblemsNotation:

• (M,π): Poisson manifold

• S ⊂M: closed Poisson submanifold

Local Model Problem:

I Is there a 1st order local model around S?

Linearization/Normal Form Problem:

I When is (M,π) locally isomorphic to the 1st order localmodel around S?

Neighborhood Equivalence Problem:

I If π ′,π are Poisson structures with same 1st order dataalong S, are they locally isomorphic?

Overview - Main Results

Theorem (Existence of local models)The class of first order jets satisfying the partial splittingcondition admits a 1st order local model.

Theorem (Linearization)If (M,π) has restriction T ∗SM integrable by a compact, Liegroupoid, whose s-fibers have trivial 2nd de Rhamcohomology, then it is linearizable at S.

Corollary (Marcut 2014)Let T ∗SM be integrable by a compact, Lie groupoid, whoses-fibers have trivial 2nd de Rham cohomology. If J1

Sπ ′ = J1Sπ,

then π ′ is locally isomorphic to π around S.

Sπ ′ = J1Sπ,

Overview - Problems - Global Version

Notation:

• (G,ω)⇒M: symplectic groupoid: ω closed, multiplicative,ker ω = 0.

• (GS ,ωS)⇒ S: over-symplectic groupoid: ωS closed,multiplicative, ker ωS ⊂ ker ds∩ker dt.

Notation:

Facts:

• Restriction of (G,ω) to saturated submanifold S ⊂M is anover-symplectic groupoid (G|S ,ω|S), and

• Inclusion i : (G|S ,ω|S) → (G,ω) is a coisotropic embedding.

Notation:

Facts:

• Restriction of (G,ω) to saturated submanifold S ⊂M is anover-symplectic groupoid (G|S ,ω|S), and

• Inclusion i : (G|S ,ω|S) → (G,ω) is a coisotropic embedding.

Notation:

Groupoid Coisotropic Embedding Problem:

I Given an over-symplectic groupoid (GS ,ωS), is there agroupoid coisotropic embedding i : (GS ,ωS) → (G,ω) intosome symplectic groupoid?

I Given groupoid coisotropic embeddings (k = 1,2)ik : (GS ,ωS) → (Gk ,ωk ), is there a local symplecticgroupoid isomorphism Φ : (G1,ω1)→ (G2,ω2), defined ina neighborhood of i1(GS), and such that Φ i1 = i2?

Overview - Main Results - Global Version

Theorem (Existence of coisotropic embeddings)If (GS ,ωS) is partially split then (ker ωS)∗⇒ (ker ωS |S)∗ is asymplectic groupoid and the zero section i : GS → (ker ωS)∗ isa coisotropic embedding.

Proper over-symplectic groupoids are partially split.

Theorem (Normal form for coisotropic embeddings)A groupoid coisotropic embedding i : (GS ,ωS) → (G,ω) into as-proper symplectic groupoid is locally isomorphic to thecoisotropic embedding i : GS → (ker ωS)∗.

Together these imply:

Groupoid Coisotropic Neighboorhood Theorem

Overview - Brief history of smooth linearization

For a fixed point S = x0:

• Weinstein 83: Linearization problem, local form, conjectures onlinearization

• Conn 85: Linearization for g compact semi-simple (Nash-Moser)

• Monnier & Zung 04: Linearization for g⊕R, when g is compactsemi-simple (Nash-Moser)

• Crainic & RLF 11: Geometric proof of Conn’s Theorem

For a symplectic leaf S:

• Vorbjev 01, 05: First order local model and some partial results onneighborhood equivalence;

• Crainic & Marcut 12: Linearization theorem for symplectic leaves(geometric proof extending Crainic & RLF 11)

For a general Poisson submanifold S:

• Marcut 14: Rigidity (Nash-Moser)

Our Linearization Theorem includes all these results, but. . .

Plan of the talk

I. First order data: jets of Poisson structures

II. Global theory: groupoid coisotropic embeddings

III. Local theory: existence of local models and linearization

I. First order data

Notation:

• (T ∗M, [·, ·]π ,π]): cotangent Lie algebroid of (M,π).

Observation: This is really about jets of bivector fields!

I. First order data

Notation:

Proposition (Coste-Dazord-Weinstein, 87)

Poisson structures

π ∈ X2(M)

←→

Lie algebroids (T ∗M, [·, ·],ρ) withρ : T ∗M → TM skew-symmetric

[Ω1cl(M),Ω1

cl(M)]⊂ Ω1cl(M)

I. First order data

Notation:

Proposition (Modern formulation)Poisson structures

π ∈ X2(M)

←→

Lie algebroids (T ∗M, [·, ·],ρ) with

id : T ∗M → T ∗M a closed IM 2-form

I. First order data

Notation:

π ∈ X2(M)

←→

A closed IM 2-form on a Lie algebroid (A, [·, ·],ρ) is a bundle mapµ : A→ T ∗M such that:

〈µ(α),ρ(β )〉=−〈µ(β ),ρ(α)〉µ([α,β ]) = Lρ(α)µ(β )− iρ(β )dµ(α)

I. First order data

Notation:

π ∈ X2(M)

←→

Notation:

• J1E →M: first jet bundle of a bundle E →M.

• The Schouten bracket induces a graded Lie bracket:

[ , ] : Γ(J1(∧k+1TM))×Γ(J1(∧l+1TM))→ Γ(J1(∧k+l+1TM)),

characterized by

[J1ϑ1,J1

ϑ2] := J1[ϑ1,ϑ2].

Proof.i : J1(∧d TM) → Derd (T ∗M),

i(J1ϑ)(α1, . . . ,αd ) := d(ϑ(α1, . . . ,αd ))−

∑i=1

(−1)i iϑ ](α1,...,αi ,...,αd )dαi ,

where for ϑ ∈ Xd (M) we denote by ϑ ] : ∧d−1T ∗M → TM the map given by

ϑ](α1, . . . ,αk−1)(α) := ϑ(α1, . . . ,αk−1,α).

This map:

(i) takes the Schouten bracket to the graded bracket of multi-derivations.

(ii) image is multi-derivations D with symbol σD ∈ ∧d T ?M ⊂ ∧d−1TM⊗TM

Notation:

characterized by

[J1ϑ1,J1

ϑ2] := J1[ϑ1,ϑ2].

Question. What is the geometric meaning of τ ∈ Γ(J1(∧2TM)) with[τ,τ] = 0?

i(J1ϑ)(α1, . . . ,αd ) := d(ϑ(α1, . . . ,αd ))−

∑i=1

(−1)i iϑ ](α1,...,αi ,...,αd )dαi ,

ϑ](α1, . . . ,αk−1)(α) := ϑ(α1, . . . ,αk−1,α).

This map:

Notation:

characterized by

[J1ϑ1,J1

ϑ2] := J1[ϑ1,ϑ2].

Propositionτ ∈ Γ(J1(∧2TM)) with

[τ,τ] = 0

←→

Holonomic jets (i.e.., Poisson structures) correspond to Liealgebroids for which id : T ∗M → T ∗M is a closed IM 2-form.

i(J1ϑ)(α1, . . . ,αd ) := d(ϑ(α1, . . . ,αd ))−

∑i=1

(−1)i iϑ ](α1,...,αi ,...,αd )dαi ,

ϑ](α1, . . . ,αk−1)(α) := ϑ(α1, . . . ,αk−1,α).

This map:

[τ,τ] = 0

←→

i(J1ϑ)(α1, . . . ,αd ) := d(ϑ(α1, . . . ,αd ))−

∑i=1

(−1)i iϑ ](α1,...,αi ,...,αd )dαi ,

ϑ](α1, . . . ,αk−1)(α) := ϑ(α1, . . . ,αk−1,α).

This map:

[τ,τ] = 0

←→

i(J1ϑ)(α1, . . . ,αd ) := d(ϑ(α1, . . . ,αd ))−

∑i=1

(−1)i iϑ ](α1,...,αi ,...,αd )dαi ,

ϑ](α1, . . . ,αk−1)(α) := ϑ(α1, . . . ,αk−1,α).

This map:

[τ,τ] = 0

←→

i(J1ϑ)(α1, . . . ,αd ) := d(ϑ(α1, . . . ,αd ))−

∑i=1

(−1)i iϑ ](α1,...,αi ,...,αd )dαi ,

ϑ](α1, . . . ,αk−1)(α) := ϑ(α1, . . . ,αk−1,α).

This map:

Notation:

• S ⊂M: closed submanifold.

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

The restriction J1(∧•TM)|S → S has the sub-bundle:

J1S(∧•TM) := τ ∈ J1(∧•TM)|S : pr∧•TSM(τ) ∈ ∧•TS.

which comes with

I projections:J1

S(∧•TM) //

J1(∧•TS)

∧•TS

I a graded Lie bracket on its sections

[ , ] : Γ(J1S(∧k+1TM))×Γ(J1

S(∧l+1TM))→ Γ(J1S(∧k+l+1TM)).

We call τ ∈ Γ(J1S(∧•TM)) holonomic if τ = (J1ϑ)|S for ϑ ∈ X•S(M)

Notation:

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

which comes with

I projections:J1

S(∧•TM) //

J1(∧•TS)

∧•TS

[ , ] : Γ(J1S(∧k+1TM))×Γ(J1

S(∧l+1TM))→ Γ(J1S(∧k+l+1TM)).

Notation:

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

which comes with

I projections:J1

S(∧•TM) //

J1(∧•TS)

∧•TS

[ , ] : Γ(J1S(∧k+1TM))×Γ(J1

S(∧l+1TM))→ Γ(J1S(∧k+l+1TM)).

Notation:

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

which comes with

I projections:J1

S(∧•TM) //

J1(∧•TS)

∧•TS

[ , ] : Γ(J1S(∧k+1TM))×Γ(J1

S(∧l+1TM))→ Γ(J1S(∧k+l+1TM)).

Notation:

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

which comes with

I projections:J1

S(∧•TM) //

J1(∧•TS)

∧•TS

[ , ] : Γ(J1S(∧k+1TM))×Γ(J1

S(∧l+1TM))→ Γ(J1S(∧k+l+1TM)).

Notation:

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

• J1X•S(M): holonomic sections of J1S(∧•TM)

There is a projection

X•S(M)J1

J1X•S(M)

X•(S)

ϑ 7→ J1Sϑ := (J1

ϑ)|S ,

which gives a canonical identification for the holonomic sections:

J1X•S(M)' X•S(M)/(I2S ·X

•(M)),

where IS ⊂ C∞(M) denotes the vanishing ideal of S.

Notation:

• X•S(M) := ϑ ∈ X•(M) : ϑ |S ∈ Γ(∧•TS)

• J1X•S(M): holonomic sections of J1S(∧•TM)

There is a projection

X•S(M)J1

J1X•S(M)

X•(S)

ϑ 7→ J1Sϑ := (J1

ϑ)|S ,

which gives a canonical identification for the holonomic sections:

J1X•S(M)' X•S(M)/(I2S ·X

•(M)),

where IS ⊂ C∞(M) denotes the vanishing ideal of S.

Question. What is the geometric meaning of τ ∈ Γ(J1S(∧2TM)) with

[τ,τ] = 0?

DefinitionAn element of the space:

2S(M) := τ ∈ J1X2

S(M) : [τ,τ] = 0,

is called a 1st order jet of a Poisson structure at S.

PropositionLet µS : T ∗SM → T ∗S, α 7→ α|TS . Then:

τ ∈ Γ(J1S(∧2TM)) with

[τ,τ] = 0

←→

(T ∗SM, [·, ·],ρ) with〈µS(α),ρ(β )〉=−〈µS(β ),ρ(α)〉

Holonomic jets τ ∈ J1X2

S(M) correspond to Lie algebroids forwhich µS is a closed IM 2-form.

2S(M) := τ ∈ J1X2

S(M) : [τ,τ] = 0,

[τ,τ] = 0

←→

Proof.Same as before, where now:

i : J1S(∧

d TM) → Derd (T ∗SM).

2S(M) := τ ∈ J1X2

S(M) : [τ,τ] = 0,

[τ,τ] = 0

←→

2S(M) := τ ∈ J1X2

S(M) : [τ,τ] = 0,

Summary of first order dataA first order jet of Poisson structure along a submanfiold S ⊂M, canbe thought as either:

I A class of bivector fields tangent to S:

τ ∈ J10X

2S(M) =

π ∈ X2S(M) : [π,π] ∈ I2

S ·X3S(M)

I2S ·X

I A Lie algebroid τ = (T ∗SM, [·, ·],ρ) for which the restrictionµS : T ∗SM → T ∗S, α 7→ α|TS , is a closed IM 2-form.

There is a commutative diagram:

Germs(Poisson(M,S))J1

wwPoisson(S)

These are the only things you need to remember!

Summary of first order dataA first order jet of Poisson structure along a submanfiold S ⊂M, canbe thought as either:

I A class of bivector fields tangent to S:

τ ∈ J10X

2S(M) =

π ∈ X2S(M) : [π,π] ∈ I2

S ·X3S(M)

I2S ·X

I A Lie algebroid τ = (T ∗SM, [·, ·],ρ) for which the restrictionµS : T ∗SM → T ∗S, α 7→ α|TS , is a closed IM 2-form.

There is a commutative diagram:

wwPoisson(S)

These are the only things you need to remember!

Example (Marcut 13)

• M = R3 with coordinates (x ,y ,z);

• S = (x ,y ,0) : x ,y ∈ R ⊂ R3 (xy -plane)

Claim 1: The bivector field π ∈ X2S(M) given by:

π = z∂x ∧∂y + xz∂x ∧∂z

defines a 1st order jet τ = J1S(π) ∈ J1

0X2S(M).

Claim 2: The 1st order jet τ is not in the image of J1S :

wwPoisson(S)

Example (Marcut 13)

• S = (x ,y ,0) : x ,y ∈ R ⊂ R3 (xy -plane)

π = z∂x ∧∂y + xz∂x ∧∂z

0X2S(M).

Proof:[π,π] = z2

∂x ∧∂y ∧∂z ∈ I2S ·X

wwPoisson(S)

Example (Marcut 13)

• S = (x ,y ,0) : x ,y ∈ R ⊂ R3 (xy -plane)

π = z∂x ∧∂y + xz∂x ∧∂z

0X2S(M).

wwPoisson(S)

Example (Marcut 13)

• S = (x ,y ,0) : x ,y ∈ R ⊂ R3 (xy -plane)

π = z∂x ∧∂y + xz∂x ∧∂z

0X2S(M).

wwPoisson(S)

Proof: Any π ∈ X2S(M) such that J1(π) = τ takes the form:

π = (z + z2f1)∂x ∧∂y + (xz + z2f2)∂x ∧∂z + z2f3∂y ∧∂z , fi ∈ C∞(R3)

=⇒ [π, π] = z2(1 + O(1))∂x ∧∂y ∧∂z 6= 0

Example (Marcut 13)

• S = (x ,y ,0) : x ,y ∈ R ⊂ R3 (xy -plane)

π = z∂x ∧∂y + xz∂x ∧∂z

0X2S(M).

wwPoisson(S)

Proof: Any π ∈ X2S(M) such that J1(π) = τ takes the form:

π = (z + z2f1)∂x ∧∂y + (xz + z2f2)∂x ∧∂z + z2f3∂y ∧∂z , fi ∈ C∞(R3)

=⇒ [π, π] = z2(1 + O(1))∂x ∧∂y ∧∂z 6= 0

First order local models:

wwPoisson(S)

DefinitionGiven a class C ⊂ J1

0X2S(M) a first order local model for C is

a splitting σ : C −→ Germs(Poisson(M,S)) of the map J1S .

Theorem (Existence of local models)The class C ⊂ J1

0X2S(M) satisfying the partially split condition

admits a 1st order local model σ : C→ Germs(Poisson(M,S)).

wwPoisson(S)

Splittings give linear models:

• if π ∈ X2S(M) is Poisson and J1

S(π) ∈ C, then σ(J1S(π)) ∈ X2

S(M)is Poisson with the same first order jet at S as π.

wwPoisson(S)

Normal forms:

wwPoisson(S)

DefinitionA local model σ : C→ Germs(Poisson(M,S)) is called a localnormal form for C if given τ ∈ C every Poisson structure π

with J1S(π) = τ is locally isomorphic to the linear model σ(τ).

Theorem (Linearization/normal form)Let C ⊂ J1

0X2S(M) be the class for which T ∗SM is integrable by a

compact, Lie groupoid whose s-fibers have trivial 2nd deRham cohomology. It is partially split and admits a localnormal form σ .

Normal forms:

wwPoisson(S)

DefinitionA local model σ : C→ Germs(Poisson(M,S)) is called a localnormal form for C if given τ ∈ C every Poisson structure π

with J1S(π) = τ is locally isomorphic to the linear model σ(τ).

Theorem (Linearization/normal form)Let C ⊂ J1

0X2S(M) be the class for which T ∗SM is integrable by a

compact, Lie groupoid whose s-fibers have trivial 2nd deRham cohomology. It is partially split and admits a localnormal form σ .

Definition (Bursztyn-Crainic-Weinstein-Zhu 04)An over-symplectic groupoid is a groupoid GS ⇒ S togetherwith a closed multiplicative 2-form ωS ∈ Ω2(GS) such that

ker ωS ⊂ ker dsS ∩ker dtS .

Given GS ⇒ S with a closed multiplicative 2-form ωS , the followingare equivalent:

(i) (GS ,ωS) is over-symplectic;

(ii) ∃ Poisson structure πS ∈ X2(S) such that tS : (GS ,ωS)→ (S,πS)is a forward Dirac map.

If these hold, ker ωS has constant rank = dimGS−2dimS and theorbits of GS coincide with the symplectic leaves of (S,πS).

Definition (Bursztyn-Crainic-Weinstein-Zhu 04)An over-symplectic groupoid is a groupoid GS ⇒ S togetherwith a closed multiplicative 2-form ωS ∈ Ω2(GS) such that

ker ωS ⊂ ker dsS ∩ker dtS .

Given GS ⇒ S with a closed multiplicative 2-form ωS , the followingare equivalent:

(i) (GS ,ωS) is over-symplectic;

(ii) ∃ Poisson structure πS ∈ X2(S) such that tS : (GS ,ωS)→ (S,πS)is a forward Dirac map.

If these hold, ker ωS has constant rank = dimGS−2dimS and theorbits of GS coincide with the symplectic leaves of (S,πS).

• Why is (GS ,ωS) called over-symplectic?

I If ker ωS is a simple foliation then one gets a symplecticgroupoid

(GS/ker ωS ,ωS)⇒ S,

which integrates the Poisson manifold (S,πS).

• Why are over-symplectic groupoids relevant for us?

I Given (G,ω)⇒ (M,π) and Poisson submanifold S ⊂M, then(G|S ,ω|S)⇒ S is an over-symplectic groupoid, coisotropicallyembedded in (G,ω).

I Given a coisotropic embedding of an over-symplectic groupoidi : (GS ,ωS) → (G,ω), the embedding of units (S,πS) → (M,π)is a Poisson map, and

I closed submanifolds S ⊂M correspond to close embeddingsand then S is saturated so GS = G|S .

• Why is (GS ,ωS) called over-symplectic?

I If ker ωS is a simple foliation then one gets a symplecticgroupoid

(GS/ker ωS ,ωS)⇒ S,

which integrates the Poisson manifold (S,πS).

• Why are over-symplectic groupoids relevant for us?

I Given (G,ω)⇒ (M,π) and Poisson submanifold S ⊂M, then(G|S ,ω|S)⇒ S is an over-symplectic groupoid, coisotropicallyembedded in (G,ω).

I Given a coisotropic embedding of an over-symplectic groupoidi : (GS ,ωS) → (G,ω), the embedding of units (S,πS) → (M,π)is a Poisson map, and

I closed submanifolds S ⊂M correspond to close embeddingsand then S is saturated so GS = G|S .

PropositionIf GS ⇒ S is a source 1-connected Lie groupoid with Liealgebroid AS → S, there is a 1:1 correspondence:

over-symplectic structuresω ∈ Ω2(GS)

←→

surjective closed IM 2-forms

µ : AS → T ∗S

Given:

• (GS ,ω)⇒ S: over-symplectic groupoid integrating (AS ,µ)

• coisotropic embedding i : (GS ,ωS) → (G,ω)

We obtain:

(i) a Poisson embedding (S,πS) → (M,πM);

(ii) an isomorphism (AS ,µ)' (T ∗SM,µS);

(iii) a solution (M,πM) of the realization problem for τ = (T ∗SM,µS):

J1SπM = τ.

PropositionIf GS ⇒ S is a source 1-connected Lie groupoid with Liealgebroid AS → S, there is a 1:1 correspondence:

over-symplectic structuresω ∈ Ω2(GS)

←→

surjective closed IM 2-forms

µ : AS → T ∗S

Given:

• (GS ,ω)⇒ S: over-symplectic groupoid integrating (AS ,µ)

• coisotropic embedding i : (GS ,ωS) → (G,ω)

We obtain:

(i) a Poisson embedding (S,πS) → (M,πM);

(ii) an isomorphism (AS ,µ)' (T ∗SM,µS);

(iii) a solution (M,πM) of the realization problem for τ = (T ∗SM,µS):

J1SπM = τ.

We can summarize this discussion by the diagram:

Germs of symplecticgroupoids (G,ω)⇒M

along a saturated S ⊂M

restrictionalong S //

over-symplecticgroupoids

(GS ,ωS)⇒ S

coisotropicembedding

symplectic groupoids(GS/kerωS ,ωS)⇒ S

.. and this is the groupoid version of the diagram:

splittingnn

yyPoisson(S)

This is all you have to remember!

Example (Transitive over-symplectic groupoids)

• principal K -bundle p : P→ S

• symplectic form η ∈ Ω2(S),

One obtains a transitive over-symplectic groupoid:

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

where q : P×P→ P×K P.

I Every transitive over-symplectic groupoid (GS ,ωS) takes thisform.

I It has a coisotropic embedding in the symplectic groupoid:

G := (P×P)×K Lie(K )∗⇒ P×K Lie(K )∗

q∗ω := (p pr2)∗η− (p pr1)∗η + d〈pr∗2 θ , ·〉−d〈pr∗1 θ , ·〉

where q : P×P×Lie(K )∗→ (P×P)×K Lie(K )∗ and θ is aprincipal bundle connection.

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

Example (with no coisotropic embeddings)

Integrating the 1st jet τ along S = R2 defined by:

π = z∂x ∧∂y + xz∂x ∧∂z ∈ X2(R3)

one obtains an over-symplectic groupoid (GS ,ωS):

• (GS ,ω)⇒ S is the bundle of groups pr : R5 = R2×R3→ R2 withmultiplication:

(x ,y ,u1,v1,w1,u2,v2,w2) 7→ (u1 +u2,v1 +exu1v2,w1 +w2 +exu1 −1

• ωS is the multiplicative closed 2-form:

ωS = dx ∧du + dy ∧dv + w dx ∧dy −x dy ∧dw .

=⇒ (GS ,ωS) does not admit a groupoid coisotropicembedding into some symplectic groupoid!

... but there are many over-symplectic bundles of Lie groups that admit coisotropicembeddings!

π = z∂x ∧∂y + xz∂x ∧∂z ∈ X2(R3)

Can’t we apply Gotay’s coisotropic embedding thm?

• (C,ωC): presymplectic manifold⇔ ωC closed with constant rank

Gotay’s normal form: A coisotropic embedding i : (C,ωC)→(X ,ω) is locally equivalent to:

(C,ωC) → ((ker ωC)∗,ω0)where:

ω0 := pr∗ωC + j∗ωcan

for some choice of splitting:

T ∗C = (Ker ω)∗⊕E =⇒ j : (ker ωC)∗ → T ∗C

Can’t we apply Gotay’s coisotropic embedding thm?• (C,ωC): presymplectic manifold⇔ ωC closed with constant rank

Idea of proof:T ∗CX

(ω−1)] //

section

// (ker ωC)∗ '// ν(C)

'tubular

neighborhood

Moser method: can choose tubular neighbhd (X ,ω)' ((ker ωC)∗,ω0)

Can’t we apply Gotay’s coisotropic embedding thm?• (C,ωC): presymplectic manifold⇔ ωC closed with constant rank

Idea of proof:T ∗CX

(ω−1)] //

section

// (ker ωC)∗ '// ν(C)

'tubular

neighborhood

Moser method: can choose tubular neighbhd (X ,ω)' ((ker ωC)∗,ω0)

What happens with (ker ωS)∗ for over-symplectic groupoid (GS ,ωS)?

I Is it a groupoid?

I Is it a symplectic groupoid?

LemmaFor an over-symplectic groupoid (GS ,ωS) with (AS ,µS):I k := (ker ωS)|S → S is a subbundle of TSGS contained in

the tangent to the isotropy groups and k = ker µS .

I ker ωS ⇒ S is a subgroupoid of TGS ⇒ TS canonicalisomorphic to the groupoid GS×S k⇒ S, where:

(g,v) · (h,w) = (gh,v + gw).

Important: ker ωS ⇒ S is a VB-groupoid:

GS n kker ωS =

LemmaFor an over-symplectic groupoid (GS ,ωS) with (AS ,µS):I k := (ker ωS)|S → S is a subbundle of TSGS contained in

the tangent to the isotropy groups and k = ker µS .

I ker ωS ⇒ S is a subgroupoid of TGS ⇒ TS canonicalisomorphic to the groupoid GS×S k⇒ S, where:

(g,v) · (h,w) = (gh,v + gw).

Important: ker ωS ⇒ S is a VB-groupoid:

GS n kker ωS =

ker ωS ⇒ S is a VB-groupoid:

GS n kker ωS =

I (ker ωS)∗ has groupoid structureI GS → (ker ωS)∗ is a closed subgroupoidI A coisotropic embedding i : (GS ,ωS) → (G,ω) yields a groupoid

isomorphism:

T ∗GSG

(ω−1)] //

(ker ωS)∗ '

// ν(GS)

Dually, we have VB groupoid (ker ωS)∗ = GS n k∗:

GS n k∗(ker ωS)∗ =

22 T ∗GSoooo

k∗(ker ωS)|∗S =

33oooo A∗S

AS ≡ Lie algebroid of GS k∗ = (ker µS)∗

isomorphism:

T ∗GSG

(ω−1)] //

(ker ωS)∗ '

// ν(GS)

Dually, we have VB groupoid (ker ωS)∗ = GS n k∗:

GS n k∗(ker ωS)∗ =

22 T ∗GSoooo

k∗(ker ωS)|∗S =

33oooo A∗S

AS ≡ Lie algebroid of GS k∗ = (ker µS)∗

isomorphism:

T ∗GSG

(ω−1)] //

(ker ωS)∗ '

// ν(GS)

What about the symplectic structure on (ker ωS)∗?

It depends on a choice of splitting:

T ∗GS = (ker ωS)∗⊕E

Need to choose VB subgroupoid E ⊂ T ∗GS (it may not be possible!).

DefinitionAn over-symplectic groupoid (GS ,ωS) is called partially split ifthere is a VB groupoid morphism

Θ : (ker ωS)∗→ T ∗GS ,

splitting the projection T ∗GS → (ker ωS)∗. One calls Θ a partialsplitting.

Symplectic groupoid local model of (GS,ωS)

Local Model:(ker ωS)∗ = GS n k∗⇒ k∗, ω0 := pr∗ωS + Θ∗ωcan

for a choice of partial splitting Θ : (ker ωS)∗→ T ∗GS :

Note: ω is multiplicative, closed, and non-degenerate in a groupoid neighborhood ofthe zero section.

Recall transitive over-symplectic groupoids take the form

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

for a principal K -bundle p : P→ (S,η). They are always partiallysplit and

partial splittingsΘ : (kerωS)∗→ T ∗GS

←→

principal bundleconnections θ

The resulting groupoid local model is exactly the one we saw before.

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

for a principal K -bundle p : P→ (S,η).

They are always partiallysplit and

←→

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

←→

GS := P×K P⇒ S,

q∗ωS := (p pr2)∗η− (p pr1)∗η

←→

Uniqueness of local models

ω0 = pr∗ωS︸︷︷︸constant

+Θ∗ωcan︸︷︷︸linear

Where with Mt : GS n k∗→GS n k∗, (g,v) 7→ (g, tv):

• ω constant⇐⇒ M∗t ω = ω, ∀t ;

• ω linear⇐⇒ M∗t ω = tω, ∀t ;

Linear, closed, multiplicative 2-forms are exact with a linearmultiplicative primitive, and this leads to:

PropositionThe local models associated with partial splittings Θ0 and Θ1are isomorphic: there exists a local groupoid automorphismΦ : GS n k∗→GS n k∗ fixing GS , such that Φ∗ω1 = ω0.

Uniqueness of local models

ω0 = pr∗ωS︸︷︷︸constant

+Θ∗ωcan︸︷︷︸linear

Where with Mt : GS n k∗→GS n k∗, (g,v) 7→ (g, tv):

• ω constant⇐⇒ M∗t ω = ω, ∀t ;

• ω linear⇐⇒ M∗t ω = tω, ∀t ;

Linear, closed, multiplicative 2-forms are exact with a linearmultiplicative primitive, and this leads to:

PropositionThe local models associated with partial splittings Θ0 and Θ1are isomorphic: there exists a local groupoid automorphismΦ : GS n k∗→GS n k∗ fixing GS , such that Φ∗ω1 = ω0.

Existence of local models

If (GS ,ωS) is over-symplectic with k := ker(ωS)|S = ker µS , tfae:

(i) (GS ,ωS) is partially split;

(ii) ∃ ω lin ∈ Ω2(GS n k∗) linear, closed, multiplicative, with:

ωlin|k×Sk

∗ = ωcan, (k×S k∗ ⊂ T (GS n k∗));

(iii) ∃ α ∈ Ω1(GS ,k) multiplicative with

α|k = id (k⊂ TSGS);

Moreover, these are equivalent to existing ω ∈ Ω2(GS n k∗) closed,multiplicative, non-degenerate at GS with:

i∗ω = ωS (i : GS →GS n k∗).

ωlin|k×Sk

∗ = ωcan, (k×S k∗ ⊂ T (GS n k∗));

ωlin|k×Sk

∗ = ωcan, (k×S k∗ ⊂ T (GS n k∗));

Proof: Given such ω, the limit:

ωlin := lim

M∗t (ω−pr∗ωS),

satisfies (ii).

ωlin|k×Sk

∗ = ωcan, (k×S k∗ ⊂ T (GS n k∗));

Example (Important!)All s-proper, over-symplectic groupoids are partially split.

Groupoid coisotropic neighborhood theorem

TheoremA symplectic groupoid (G,ω) which is invariantly linearizablearound a saturated S ⊂M is partially split and locallyisomorphic to the symplectic groupoid local model of (GS ,ωS).

Note: No properness assumption!

CorollaryA s-proper symplectic groupoid (G,ω) is locally isomorphicaround a saturated S ⊂M to the symplectic groupoid localmodel of (GS ,ωS).

Groupoid coisotropic neighborhood theorem

TheoremA symplectic groupoid (G,ω) which is invariantly linearizablearound a saturated S ⊂M is partially split and locallyisomorphic to the symplectic groupoid local model of (GS ,ωS).

Note: No properness assumption!

CorollaryA s-proper symplectic groupoid (G,ω) is locally isomorphicaround a saturated S ⊂M to the symplectic groupoid localmodel of (GS ,ωS).

III. Local theory

Philosophical Principle:

Poisson Geometry = Infinitesimal MultiplicativeSymplectic Geometry

This leads to a Dictionary.

To figure out its entries, one only has to recall the correspondences:Lie groupoids

←→

Lie algebroids

multiplicative forms(with coefficients)

←→

IM forms

(with coefficients)

multiplicativevector fields

←→

Lie algebroidderivations

III. Local theory

←→

Lie algebroids

←→

IM forms

(with coefficients)

←→

III. Local theory

←→

Lie algebroids

←→

IM forms

(with coefficients)

←→

Dictionary

Global side Infinitesimal side

symplectic groupoid(G,ω)⇒M

Poisson manifold (M,π)⇔ algebroid T ∗M such that

id : T ∗M → T ∗M is closed IM-form

saturated submanifoldS ⊂M

Poisson submanifoldS ⊂M

over-symplectic groupoid(GS ,ωS)⇒ S

1st order local data τ ∈ J1SX0(M)

⇔ algebroid T ∗SM such thatµS : T ∗SM → T ∗S is closed IM-form

coisotropic embedding(GS ,ωS) → (G,ω)

S ⊂ (M,π) a realizationof τ = (T ∗SM,µS) with τ = J1

partial splittingΘ : GS n k∗→ T ∗GS

partial splitting for T ∗SM

local normal(GS n k∗,ω0)

local model for τ = (T ∗SM,µS)

Local model - algebroid aspects

• τ = (T ∗SM,µS): 1st order local data

• k = ker µS → S: it is a representation of T ∗SM with ∇αs = [α,s]

So we can form the action algebroid:

T ∗SM n k∗→ k∗

PropositionIf τ = J1

Sπ for a Poisson structure (M,π), then k = ν(S)∗ and

T ∗SM n k∗ ' T ∗SM nν(S)

is the linear approximation of the Lie algebroid T ∗M along thesaturated submanifold S ⊂M.

Local model - algebroid aspects

• τ = (T ∗SM,µS): 1st order local data

• k = ker µS → S: it is a representation of T ∗SM with ∇αs = [α,s]

So we can form the action algebroid:

T ∗SM n k∗→ k∗

PropositionIf τ = J1

Sπ for a Poisson structure (M,π), then k = ν(S)∗ and

T ∗SM n k∗ ' T ∗SM nν(S)

is the linear approximation of the Lie algebroid T ∗M along thesaturated submanifold S ⊂M.

Local model - Poisson geometric aspectsNote: In general, T ∗SM n k∗→ k∗ is not the cotangent algebroid of aPoisson structure on k∗.

DefinitionA pair (T ∗SM,µS) is called partially split if there exists a Liealgebroid morphism

θ : T ∗SM n k∗→ T ∗(T ∗SM)

which is a splitting of the natural projection

p : T ∗(T ∗SM)→ T ∗SM n k∗.

We call θ a partial splitting.

Note: For a Lie algebroid A→M we have its cotangent algebroid:

T ∗A→ A∗,

It comes with a canonical, non-degenerate, linear, closed, IM 2-form:

µcan : T ∗A→ T ∗A∗.

This is commonly known as the Tulczyjew isomorphism.

θ : T ∗SM n k∗→ T ∗(T ∗SM)

p : T ∗(T ∗SM)→ T ∗SM n k∗.

T ∗A→ A∗,

θ : T ∗SM n k∗→ T ∗(T ∗SM)

p : T ∗(T ∗SM)→ T ∗SM n k∗.

T ∗A→ A∗,

Local modelIf (T ∗SM,µS) is partially split, a splitting θ : T ∗SM n k∗→ T ∗(T ∗SM) yieldsa closed IM 2-form:

µ0 = pr∗ µS︸︷︷︸constant

+θ∗µcan︸︷︷︸

linear

Definition

(T ∗SM n k∗,µ0).

is the local model of the partially split jet τ = (T ∗SM,µS),associated with the splitting θ .

I The form µ0 is non-degenerate in a neighborhood U of the zerosection S → T ∗SM n k∗.

I In U, we get a local model Poisson structure:

π0 = ρ µ−10 .

I (U,π0) has 1st order jet at S the local data: J1Sπ0 = τ.

linear

Definition

(T ∗SM n k∗,µ0).

π0 = ρ µ−10 .

linear

Definition

(T ∗SM n k∗,µ0).

π0 = ρ µ−10 .

If (GS ,ωS) is over-symplectic with k := ker(ωS)|S = ker µ, tfae:

ωlin|k×Sk

∗ = ωcan, (k×S k∗ ⊂ T (GS n k∗));

If τ = (T ∗SM,µS) is first order local data, k = ker µS , tfae:

(i) (T ∗SM,µS) is partially split;

(ii) ∃ linear, closed, IM form µ lin ∈ Ω2IM(T ∗SM) such that

µlin|S : T ∗SM → T ∗S⊕ k, (prk µ lin|S)|k = id

(iii) ∃ k-valued, IM form D ∈ Ω1(T ∗SM,k) whose symboll : T ∗SM → k satisfies l |k = id.

Moreover, these are equivalent to existing a closed, IM 2-formµ ∈ Ω2

IM(T ∗SM) satisfying:

µ|S : T ∗S⊕ k→ T ∗S⊕ k, prT ∗S µ|S = µS , (prk µ|S)|k = id

Example (Transitive algebroids are split)

0 // k // T ∗SMρ //

hh TS //τ

We obtain a linear operator D : Γ(T ∗SM)→Ω1(S,k) satisfying (iii):

D(α)(X) := l([τ(X),α])

The corresponding local model is precisely Vorobjev’s local model.

0 // k // T ∗SMρ //

hh TS //τ

D(α)(X) := l([τ(X),α])

0 // k // T ∗SMρ //

hh TS //τ

D(α)(X) := l([τ(X),α])

Uniqueness of the local model

PropositionLet (T ∗SM,µS) be a partially split. Given splittings θ0 and θ1,there is an algebroid automorphism Φ : T ∗SM n k∗→ T ∗SM n k∗

defined around S such that:

Φ|T ∗SM = id, Φ∗µ0 = µ1.

Proof: Every linear, closed, IM 2-form is exact. So apply a IM versionof Moser to the path:

µt = µS + tµlin1 + (1− t)µ

lin0 ,

Instead of the flow of a vector field, use the flow of a time-dependentLie algebroid derivation.

Uniqueness of the local model

PropositionLet (T ∗SM,µS) be a partially split. Given splittings θ0 and θ1,there is an algebroid automorphism Φ : T ∗SM n k∗→ T ∗SM n k∗

defined around S such that:

Φ|T ∗SM = id, Φ∗µ0 = µ1.

Proof: Every linear, closed, IM 2-form is exact. So apply a IM versionof Moser to the path:

µt = µS + tµlin1 + (1− t)µ

lin0 ,

Instead of the flow of a vector field, use the flow of a time-dependentLie algebroid derivation.

Linearization Theorem

TheoremLet τ = (T ∗SM,µS) be a first order jet of a Poisson structure. IfT ∗SM is integrable by a compact Lie groupoid whose s-fibershave trivial 2nd de Rham cohomology, then:

(i) (T ∗SM,µS) is partially split, and

(ii) any (M,π) with J1Sπ = τ is locally isomorphic to the local

model.

Assume: ∃ integration (GS ,ωS)⇒ S compact, has 1-connecteds-fibers with trivial 2nd de Rham cohomology.

Geometric Proof: (GS ,ωS) It is partially split, so (i) follows. Theproof of (ii) consists of two steps:

1) Show that a neighborhood of S in (M,π) is integrable by ans-proper symplectic groupoid (G,ω);

2) Apply the groupoid coisotropic neighborhood theorem to theembedding (GS ,ωS) → (G,ω).

model.

Nash-Moser Proof: Use [Marcut 2014] and apply groupoidcoisotropic normal form.

model.

Linearization Thm. - Comments on step 1 of the proof1) Show that a neighborhood of S in (M,π) is integrable by an

s-proper symplectic groupoid (G,ω);

This step uses:

(i) The path space approach to integrability;

(ii) A result from Crainic & RLF, “A geometric approach to Conn’slinearization theorem”.

(iii) The same result was already used by Crainic & Marcut, for theirtheorem of linearization around symplectic leaves.

PropositionLet F be a foliation of finite codimension on a Banach manifold X and letY ⊂ X be a submanifold which is saturated with respect to F (i.e., each leaf ofF which hits Y is contained in Y ). Assume that:

(H0) The holonomy of F at all points in Y is trivial.

(H1) The foliation FY := F |Y is simple, i.e. its leaves are the fibers of asubmersion p : Y → B into a compact manifold B.

(H2) The fibration p : Y → B is locally trivial.

Then one can find:

(i) a transversal TX ⊂ X to the foliation F such that TY := Y ∩TX is acomplete transversal to FY (i.e., intersects each leaf of FY at leastonce).

(ii) a retraction r : TX → TY .

(iii) an action of the holonomy of FY on r : TX → TY along F .

Moreover, the orbit space TX /HolTY(FY ) is a smooth (Hausdorff) manifold.

Linearization Thm. - Comments on step 1 of the proof

PropositionLet F be a foliation of finite codimension on a Banach manifold X and letY ⊂ X be a submanifold which is saturated with respect to F (i.e., each leaf ofF which hits Y is contained in Y ). Assume that:

(H0) The holonomy of F at all points in Y is trivial.

(H1) The foliation FY := F |Y is simple, i.e. its leaves are the fibers of asubmersion p : Y → B into a compact manifold B.

(H2) The fibration p : Y → B is locally trivial.

Then one can find:

(i) a transversal TX ⊂ X to the foliation F such that TY := Y ∩TX is acomplete transversal to FY (i.e., intersects each leaf of FY at leastonce).

(ii) a retraction r : TX → TY .

(iii) an action of the holonomy of FY on r : TX → TY along F .

Moreover, the orbit space TX /HolTY(FY ) is a smooth (Hausdorff) manifold.

Existence of other normal forms

Local normal forms were defined as splittings:

splittingnn

yyPoisson(S)

I We described a normal form for partially split local data;

I For b-symplectic manifolds, the normal form around the singularlocus is a different kind of normal form!

Are there other kinds of local normal forms?

Rui Loja Fernandes (based on joint work with Ioan Marcut˘2020/04/09 · Local models around...

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