The modular class of a Poisson map

TL;DR

This paper introduces the modular class of a Poisson map, exploring its properties, examples, and implications for Poisson manifold reduction and symplectic groupoids, advancing understanding of Poisson geometry.

Contribution

It defines the modular class of a Poisson map and studies its behavior and applications, including in reduction and symplectic groupoid contexts.

Findings

Modular class of a Poisson map is well-defined and useful.

Behavior of modular classes under reduction is characterized.

Connections to symplectic groupoid cohomology are established.

Abstract

We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.