Symplectic and Poisson geometry on b-manifolds

TL;DR

This paper systematically studies b-Poisson manifolds, focusing on their local structure near the singular set, and introduces topological invariants and a variant of de Rham theory to understand their geometry and Poisson cohomology.

Contribution

It provides a detailed analysis of b-Poisson manifolds' structure near the singular set and develops new topological invariants and a modified de Rham theory for these manifolds.

Findings

Characterization of local structure near the singular set Z

Introduction of topological invariants for b-Poisson manifolds

Development of a variant of de Rham theory and its relation to Poisson cohomology

Abstract

Let $M^{2n}$ be a Poisson manifold with Poisson bivector field $\Pi$. We say that $M$ is b-Poisson if the map $\Pi^n:M\to\Lambda^{2n}(TM)$ intersects the zero section transversally on a codimension one submanifold $Z\subset M$. This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of $(M,\Pi)$ in the neighbourhood of $Z$ and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.