Poisson Cohomology of holomorphic toric Poisson manifolds. I

TL;DR

This paper computes the Poisson cohomology groups for all holomorphic toric Poisson structures on complex projective spaces and affine spaces, providing a comprehensive understanding of their cohomological properties.

Contribution

It offers the first complete calculations of Poisson cohomology for holomorphic toric Poisson structures on $CP^n$ and $C^n$, including algebraic and formal cases.

Findings

Poisson cohomology groups for all structures on $CP^n$ are computed.

Algebraic and formal Poisson cohomology groups on $C^n$ are determined.

Results generalize known cases and provide new insights into toric Poisson geometry.

Abstract

A holomorphic toric Poisson manifold is a nonsingular toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we computed the Poisson cohomology groups for all holomorphic toric Poisson structures on $CP^n$, with the stand Poisson structure on $CP^n$ as a special case. We also computed the algebraic and the formal Poisson cohomology groups of holomorphic toric Poisson structures on $C^n$.