# Unimodularity Criteria for Poisson Structures on Foliated Manifolds

## TL;DR

This paper investigates unimodularity criteria for Poisson structures on foliated manifolds, generalizing known conditions and establishing the importance of transverse unimodularity for local properties.

## Contribution

It introduces new unimodularity criteria in the semilocal setting around symplectic leaves and relates modular classes to the Reeb class in Poisson foliations.

## Key findings

- Unimodularity of the transverse Poisson structure is necessary for semilocal unimodularity.
- Provides an explicit formula for modular vector fields in coupling Poisson structures.
- Connects the modular class of a Poisson foliation with the Reeb class.

## Abstract

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.