# Rational Morita equivalence for holomorphic Poisson modules

## TL;DR

This paper introduces a birational Morita equivalence concept for Poisson modules on complex varieties, classifies their types, and explores the geometric structures they induce, including foliations and projective structures.

## Contribution

It defines a weak Morita equivalence for Poisson modules and classifies modules on varieties with mild singularities, linking them to flat sheaves, meromorphic connections, or co-Higgs sheaves.

## Key findings

- Poisson modules are either rationally Morita equivalent to flat, meromorphic, or co-Higgs sheaves.
- The study of rank two $rak{sl}_2$-Poisson modules reveals structures analogous to transversally projective foliations.
- The geometry of symplectic foliations induced by Poisson connections is characterized.

## Abstract

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foliations. Moreover, we describe the geometry of the symplectic foliation induced by the Poisson connection on the projectivization of the Poisson module.