Riemann-Poisson and K\"ahler-Poisson complex manifolds and structures

TL;DR

This paper explores complex Poisson brackets on smooth functions over complex manifolds, examining their compatibility with Riemannian structures and providing an example in  .

Contribution

It introduces a new class of complex Poisson structures generated by a specific closed (1,1)-form and analyzes their compatibility with Riemannian geometry.

Findings

Compatibility conditions between complex Poisson and Riemannian structures are established.

An explicit example of such structures is constructed in  .

The framework extends understanding of complex Poisson geometry in complex manifolds.

Abstract

In this paper we consider structures of complex Poisson brackets on the space of smooth functions in a $n$-dimensional complex manifold generated by the $(1,1)$-form $d=\partial+\overline{\partial}$-closed and non-degenerate (with non-holomorphic and non-antiholomorphic coefficients). In this case, we have view the compatibility between complex Poisson and Riemannian structures and an example is giving in $\C^\ast$.