# Poisson structures of near-symplectic manifolds and their cohomology

## TL;DR

This paper establishes a connection between Poisson and near-symplectic geometry by constructing a singular Poisson structure on near-symplectic 4-manifolds and analyzing its cohomology, revealing finite-dimensionality and relations to contact structures.

## Contribution

It introduces a specific singular Poisson structure on near-symplectic 4-manifolds and computes its Poisson cohomology, linking it to modular vector fields and contact geometry.

## Key findings

- Poisson structure of maximal rank 4 on near-symplectic 4-manifolds
- Finite-dimensional Poisson cohomology depending on modular vector field
- Relation between Poisson structure and overtwisted contact structures

## Abstract

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 $\pi$ is defined on the tubular neighbourhood of the singular locus $Z_{\omega}$ of the 2-form $\omega$, it is of maximal rank 4 and it vanishes on a degeneracy set containing $Z_{\omega}$. 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 $\pi$ and the overtwisted contact structure associated to a near-symplectic 4-manifold.