Compatible Poisson Structures on Fibered $5$-Manifolds

TL;DR

This paper investigates a special class of Poisson structures on 5-dimensional fibered manifolds, analyzing their global behavior, singularities, and transformations that preserve their compatibility with the fiber bundle structure.

Contribution

It provides a comprehensive description of almost coupling Poisson tensors on 5-manifolds with 2-dimensional bases, including criteria for unimodularity and gauge transformations.

Findings

Characterization of singularities of Poisson tensors

Unimodularity criteria for almost coupling structures

Identification of gauge transformations preserving the structure

Abstract

We study a class of Poisson tensors on a fibered manifold which are compatible with the fiber bundle structure by the so-called almost coupling condition. In the case of a $5$-dimensional orientable fibered manifolds with $2$-dimensional bases, we describe a global behavior of almost coupling Poisson tensors and their singularities by using a bigraded factorization of the Jacobi identity. In particular, we present some unimodularity criteria and describe a class of gauge type transformations preserving the almost coupling property.