Holonomy on Poisson Manifolds and the Modular Class

TL;DR

This paper introduces linear holonomy for Poisson manifolds, linking it to the modular class, and proves modular class invariance under Morita equivalence for locally unimodular cases.

Contribution

It defines linear holonomy for singular Poisson structures and establishes its relation to the modular class, also proving modular class invariance under Morita equivalence.

Findings

Linear holonomy generalizes the linearized holonomy on regular symplectic foliations.

The logarithm of the determinant of linear holonomy equals the integral of the modular vector field.

Modular class is invariant under Morita equivalence for locally unimodular Poisson manifolds.

Abstract

We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The linear holonomy is closely related to the modular class studied by A. Weinstein. Namely, the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field along such a lift. This assertion relies on the notion of the integral of a vector field along a cotangent path on a Poisson manifold, which is also introduced in the paper. In the second part of the paper we prove that for locally unimodular Poisson manifolds the modular class is an invariant of Morita equivalence.