On the splitting of infinitesimal Poisson automorphisms around symplectic leaves

TL;DR

This paper provides a geometric framework for understanding the first Poisson cohomology groups near symplectic leaves, using splitting theorems for infinitesimal automorphisms to analyze the structure of Poisson manifolds.

Contribution

It introduces splitting theorems for infinitesimal automorphisms of coupling Poisson structures, revealing how tangential and transversal data interact around symplectic leaves.

Findings

Criteria for vanishing first Poisson cohomology groups

Splitting formulas applied to singular symplectic foliations

Analysis of infinitesimal automorphisms in Poisson geometry

Abstract

A geometric description of the first Poisson cohomology groups is given in the semilocal context, around (possibly singular) symplectic leaves. This result is based on the splitting theorems for infinitesimal automorphisms of coupling Poisson structures which describe the interaction between the tangential and transversal data of the characteristic distributions. As a consequence, we derive some criteria of vanishing of the first Poisson cohomology groups and apply the general splitting formulas to some particular classes of Poisson structures associated with singular symplectic foliations.