On the geometry of complex Poisson bivectors

TL;DR

This paper explores the geometric structure of complex Poisson bivectors, introduces new classes of structures, and provides a normal form theorem under regularity conditions, advancing understanding of complex Poisson geometry.

Contribution

It introduces quasi-real Poisson and Dirac structures and offers a normal form theorem for complex Poisson bivectors under regularity assumptions.

Findings

Complex Poisson bivectors have associated complex presymplectic foliations.

Introduction of quasi-real Poisson and Dirac structures.

Normal form theorem for complex Poisson structures along certain submanifolds.

Abstract

We study the geometry of complex Poisson bivectors over smooth manifolds. We show that under mild regularity conditions any complex Poisson bivector has associated a complex presymplectic foliation. After that, we use techniques of Dirac geometry to provide a more concise description of this complex presymplectic foliation. Moreover, we introduce two new classes of structures: quasi-real Poisson and quasi-real Dirac structures. In the last part, we focus on the normal form of complex Poisson bivectors. Under certain regularity, we provide a normal form theorem for complex Poisson structures along certain kinds of submanifolds.