Poisson structures of near-symplectic manifolds and their cohomology

Abstract

We connect Poisson and near-symplectic geometry by showing that there is an almost regular Poisson structure induced by a near-symplectic form ω when its singular locus is a symplectic mapping torus. This condition is automatically satisfied on any near-symplectic 4-manifold. The Poisson structure π is of maximal rank 2n and it drops its rank by 4 on a degeneracy set that coincides with the singular locus of the near-symplectic form. We then compute its Poisson cohomology in dimension 4. The cohomology spaces are finite dimensional and depend on the modular class. We conclude with a comment on the interaction between the Poisson structure π and an overtwisted contact structure.

Poisson Cohomology of Broken Lefschetz Fibrations

“…Since d 3 is a quadratic operator, no constant or linear terms are in Im(d 3 ). The kernel of d 4 being V , together with direct computation of the image d 3 (X 1 ), X 1 ∈ X 3 1 (U C ), proves the claim.…”

“…Let A = R[x 1 , x 2 , x 3 ] and φ = 1 2 Q 2 . The restriction of π Γ on A is then determined by φ in the sense that {x σ(i) , x σ(j) } = ∂ σ(k) φ for every cyclic permutation σ of (1,2,3). Denote this Poisson algebra by (A, π φ ).…”

“…The Poisson structure on a bLf, together with the Poisson structure on nearsymplectic 4-manifolds studied in [3] and log-symplectic structures on 4-manifolds, are examples of singular Poisson structures for any possible combination of degeneracies in the rank of a Poisson structure. The Poisson bivector associated to a bLF is of rank 2 or 0, whereas on a 4-manifold log-symplectic structures are Poisson structures of rank 4 or 2, and near-symplectic manifolds have a Poisson bivector of rank 4 or 0.…”

“…The Poisson bivector associated to a bLF is of rank 2 or 0, whereas on a 4-manifold log-symplectic structures are Poisson structures of rank 4 or 2, and near-symplectic manifolds have a Poisson bivector of rank 4 or 0. Hence, together with the Poisson cohomology of near-symplectic manifolds [3] and the Poisson cohomology of log-symplectic manifolds [9,16], here we complete the Poisson cohomology computation for these structures.…”