COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD
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1 COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD MELINDA LANIUS 1. introduction Because Poisson cohomology is quite challenging to compute, there are only very select cases where the answer is known. In the case of a symplectic manifold where the Poisson bi-vector is non-degenerate, the Poisson cohomology is isomorphic to the de Rham cohomology. The non-degeneracy of π allows us to define an isomorphism T M T M that provides this isomorphism in cohomology: H p (M) H p π(m). A similar story is true in the b-poisson setting, a specific case of mildly degenerate Poisson manifold. Given a b-poisson manifold, the Poisson cohomology H p π(m) us computable as H P (M) H p 1 (M). In this note we will discuss two approaches to this computation. 2. Background Given a manifold M with hypersurface Z M, the b-tangent bundle is the vector bundle whose space of sections is D = {u C (M; T M) : u Z C (Z; T Z)}. If x, y 1,..., y n are local coordinates near a point in Z, and x is a defining function for Z, then the vector fields and co-vectors x x, dx,..., y 1 y n x, dy 1,..., dy n respectively form local bases for b T M and b T M. Recall that a Lie algebroid is a triple (A, [, ] A, ρ A ) that is a triple of a vector bundle A M, a Lie bracket [, ] A on the module of sections Γ(A), and a bundle map ρ A : A T M such that [X, fy ] = L ρa (X)f Y + f[x, Y ] where X, Y Γ(A), f C (M). The b-tangent bundle can be considered a Lie algebroid with anchor map ρ the inclusion into the tangent bundle and bracket induced by the standard Lie bracket on T M. A b-poisson structure (M, Z, π) is a manifold M, a hypersurface Z M, and a Poisson bi-vector π such that there exists a nondegenerate section π Γ( 2 ( b T M)) such that ρ( π) = π. 1
2 2 MELINDA LANIUS Recall for a general Poisson manifold (M, π), the Poisson cohomology H π(m) is defined as the cohomology groups of the Lichnerowicz complex: This complex is formed using V k (M) := C (M; k T M), smooth multivectorfields on M. V k 1 (M) dπ V k (M) dπ V k+1 (M)... The differential d π : V k (M) V k+1 (M) is defined as d π = [π, ], where [, ] is the Schouten bracket extending the standard Lie bracket on vectorfields V 1 (M). 3. Established Approach In the literature, the strategy for computing the Poisson cohomology of a b-poisson manifold involves establishing isomorphisms with several more tractable complexes. We can define the b-poisson cohomology of a b-poisson manifold (M, Z, π). Let b V k (M) denote C (M; k ( b T M)), the smooth b-multivertorfields on M. The operator d π = [π, ] is a differential on this subalgebra. The b-poisson cohomology b H π(m) is the cohomology of the complex b V k 1 (M) dπ b V k (M) dπ b V k+1 (M)... We can use algebroid structures to compute a cohomology. Recall, for any Lie algebroid (A, [, ] A, ρ A ) over M, with dual A, the degree k A-forms are A Ω k (M) = Γ( k A ), the sections of the kth exterior power of the dual bundle A. The differential operator d A acting on A Ω (M), d A : A Ω k (M) A Ω k+1 (M) is defined by (d A β)(α 0, α 1,..., α k ) = + 0 i<j k k ( 1) i ρ A (α i ) β(α 0,..., ˆα i,..., α k ) i=0 ( 1) i+j β([α i, α j ] A, α 0,..., ˆα i,..., ˆα j,..., α k ) for β A Ω k (M), and α 0,..., α k Γ(A). This is a complex whose cohomology is called the Lie algebroid cohomology or A-de Rham cohomology. Victor Guillemin, Eva Miranda, and Ana Rita Pires [2], utilizing a lemma provided by Ioan Mărcut and Boris Osorno Torres in [4], show that the b-symplectic form π 1 of a non-degenerate b-poisson manifold
3 COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD 3 (M, Z, π) gives an isomorphism b T M b T M which gives an isomorphism between the b-de Rham cohomology and the b-poisson cohomology. And in fact, the b-poisson cohomology of a non-degenerate b- Poisson manifold is isomorphic to the Poisson cohomology: b H p (M) b Hπ(M) p Hπ(M). p We will discuss the three key propositions to this approach. The following is a result of Rafe Mazzeo and Richard Melrose. Proposition 3.1. Let (M, Z) be a manifold with hypersurface Z. Then, the b-de Rham cohomology b H (M) is computable as H p (M) H p 1 (Z). Proof. (sketch) we define the complex C p = b Ω p (M)/Ω p (M) the quotient under the inclusion map i : b T M T M. We have a differential C d induced by the differential b d on b Ω p (M). In particular, if π is the projection b Ω p (M) b Ω p (M)/Ω p (M), then C d(η) = π( b d(θ)) where θ b Ω p (M) is any form such that π(θ) = η. We have a short exact sequence 0 Ω p (M) i b Ω p (M) π C p 0 where i : b T M T M is the inclusion. Our short exact sequence gives us a long exact sequence in cohomology, whose boundary map δ : H p (C) H p+1 (M) is zero. Thus b H p (M) H p (M) H p (C) and we are left to compute H p (C). Let x be any Z defining function. Then any form in C p is expressible as dx ( ) dx x α for α Ωp 1 (Z). Then d x α = dx dα. Thus x kernel d : C p C p+1 {α Ω p (Z) dα = 0}. Notice that there are elements dx ( ) dx x β Cp 1 such that d x β = dx x dβ Cp. Thus the image d : C p 1 C p {α Ω p (Z) α = dβ, β Ω p 2 (Z)}. This computation is invariant under change of Z defining function. Thus H p (C) H p 1 (Z). Victor Guillemin, Eva Miranda, and Ana Rita Pires re-contextualized this above result and proved the following. Proposition 3.2. Let (M, Z, π) be a b-poisson manifold. Then, the b-poisson cohomology b H π(m) is isomorphic to the b-de Rham cohomology b H (M).
4 4 MELINDA LANIUS The details of this proof are identical to the standard symplectic case which can be found in Section of [1], and can be found in the proof of Theorem 30 in [2]. Thus, by the previous proposition, b Hπ(M) p H p (M) H p 1 (Z). The final key isomorphism comes from Ioan Mărcut and Boris Osorno Torres. Proposition 3.3. The inclusion b V k (M) V k (M) induces an isomorphism in cohomology. In their proof, Mărcut and Osorno Torres explicitly construct a inverse to the map induced by the inclusion in cohomology. 4. A new approach: Resolving algebroid In the following section, we construct an algebroid that more directly computes the Poisson cohomology. Let (M, Z, π) be a non-degenerate b-poisson manifold and let ω = π 1 be the corresponding b-symplectic form. Guillemin, Miranda, and Pires [2] showed that ω induces a cosymplectic structure (θ, η) Ω 1 (Z) Ω 2 (Z) on Z. That is, there exists a pair of closed forms such that θ η n 1 0 where the dimension of Z is 2n 1. Let R be the Reeb vectorfield associated to (θ, η). That is the non-vanishing vectorfield R on Z such that θ(r) = 1 and i R η = 0. Then T Z splits as R R R ker θ. The b resolving algebroid R is defined as the algebroid whose space of sections is {u C (M; T M) : u Z C (M; ker θ)}. For local coordinates (r, s 1, t 1,... s m, t m ) in Z such that θ = dr and η = m i=1 ds i dt i, the local sections of R are smooth linear combinations of x x, x r, s 1, t 1,..., s m, t m. Notice that we can locally identify the dual elements of R as dx x, θ x, ds 1, dt 1,... ds m, dt m. Lemma 4.1. The Poisson cohomology of a non-degenerate b-poisson manifold (M, Z, π) is isomorphic to the de Rham cohomology R H (M) of the b resolving algebroid R. Proof. Let ω be the b-symplectic form dual to π. We define a map ω : T M R by v i v ω. For all p M \ Z, R p T p M and ω p is a symplectic form. Thus for all p M \ Z, we have that ω p is an
5 COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD 5 isomorphism. For p Z, it follows from ω being non-degenerate as a scattering 2-form that ω is injective. One can verify that ω is surjective. Thus the map ω is a bundle isomorpism. By taking exterior powers of the map ω, we can extend it to an isomorphism ω : p T M p (R ) and hence a C (M)-linear isomorphism ω : V p (M) R Ω p (M). For any smooth multivectorfield η on a given smooth non-degenerate scattering Poisson manifold (M, Z, π), we have ω(d π (η)) = d( ω(η). Proof. We will proceed by induction on the degree of η and by using the Leibniz rule. Let η be a degree 0 form, that is η C (M). Then ω(η) = η and d( ω(η)) = dη. Consider d π (η) = [π, η] = X η, the Hamiltonian vectorfield of η. Thus ω( X η ) = i Xη ω = dη. If η = d π f is an exact 1-vectorfield, then ω(d π (d π f)) = ω(0) = 0 and d( ω(d π f) = d( ω(x f )) = d(df) = 0. By the Leibniz rule, the statement is true for all multivectorfields. Thus the claim shows that, up to a sign, the map ω intertwines the differential operator d of the resolving algebroid R de Rham complex with the differential operator d π of the Lichnerowicz complex. Hence ω : Hπ(M) p R H p (M) is an isomorphism. Theorem 4.2. Given a 2n-dimensional non-degenerate b-poisson manifold (M, Z, π), the Poisson cohomology H p π(m) is computable as H p π(m) H p (M) H p 1 (Z). Proof. (sketch) We are left to compute R H p (M). Similar to the computation of b-de Rham cohomology, R H p (M) H p (M) H p (C) where C p = R Ω p (M)/Ω p (M) with differential C d induced by the differential R d on R Ω p (M). In particular, if π is the projection R Ω p (M) R Ω p (M)/Ω p (M), then C d(η) = π( R d(θ)) where θ R Ω p (M) is any form such that π(θ) = η. Let x be a Z defining function. An element of C k is of the form ν = dx x θ α + dx 2 x β + θ x γ where α Ωk 2 (Z), β Ω k 1 (Z), and γ Ω k 1 (Z) such that i R α = 0 and i R γ = 0. Then dν = dx x θ 2 dα dx dx dβ x x θ γ θ 2 x dγ.
6 6 MELINDA LANIUS This gives us kernel relations γ = dα and dβ = 0. We have the element θ ( ) θ x α in Ck 1 such that d x α = dx x θ α θ 2 x dα and elements dx ( ) dx x ɛ in Ck 1 such that d x ɛ = dx dɛ. Thus x our relations give us a factor of H p 1 (Z) in cohomology. Next we will show that the class represented by β is well-defined under change of Z defining function. Let x be any Z defining function, then x = φx for some positive function φ C (M). Then d x x = dφ φ + dx x. Because φ is always positive, the dφ term does not appear φ in the projection C 1. Thus, this cohomology is invariant under change of Z defining function. 5. A comparison of methods The obvious benefit of the resolving algebroid approach is its directness. It is a shorter computation. However, information is lost. By understanding the isomorphisms in the established method, one can for instance understand deformations of b-poisson structures in the b-category versus in the category of all Poisson structures. References [1] Jean-Paul Dufour and Nguyen Tien Zung. Poisson Structures and Their Normal Forms. Progress in Mathematics, Volume 242, Birkhauser, [2] Victor Guillemin, Eva Miranda, and Ana Rita Pires. Symplectic and Poisson geometry on b-manifolds. Adv. Math. 264 (2014), [3] Victor Guillemin, Eva Miranda, Ana Rita Pires, and Geoffrey Scott. Toric actions on b-symplectic manifolds. Int Math Res Notices. first published online July 8, [4] Ioan Mărcut, and Boris Osorno Torres. Deformations of log-symplectic structures. J. Lond. Math. Soc. (2) 90 (2014), no. 1, [5]. On cohomological obstructions to the existence of log symplectic structures preprint arxiv: [6] Rafe Mazzeo. The Hodge cohomology of a conformally compact metric. J. Diff. Geom. 28 (1988), [7] Rafe Mazzeo and Richard Melrose. Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature. J. Funct. Anal. 108 (1987), Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801, USA. address: lanius2@illinois.edu
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