Poisson structures of near-symplectic manifolds and their cohomology

Journal of Mathematical Physics, Volume 62, Issue 3, March 2021. We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure π is defined on the tubular neighborhood of the singular locus Zω of the 2-form ω; it is of maximal rank 4, and it vanishes on a degeneracy set containing Zω. We compute its smooth Poisson cohomology, which depends on the modular vector field, and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure π and the overtwisted contact structure associated with a near-symplectic 4-manifold.