Poisson Cohomology of holomorphic toric Poisson manifolds. II

TL;DR

This paper describes holomorphic multi-vector fields on smooth compact toric varieties and computes their Poisson cohomology groups, extending previous results on holomorphic vector fields to more general structures.

Contribution

It generalizes Demazure's result to multi-vector fields and provides explicit computations of Poisson cohomology for toric varieties with invariant Poisson structures.

Findings

Holomorphic multi-vector fields are characterized on smooth compact toric varieties.

Poisson cohomology groups are explicitly computed for holomorphic toric Poisson manifolds.

The results extend known vector field descriptions to higher multi-vector fields.

Abstract

In this paper, we give a description of holomorphic multi-vector fields on smooth compact toric varieties, which generalizes Demazure's result of holomorphic vector fields on toric varieties. Based on the result, we compute the Poisson cohomology groups of holomorphic toric Poisson manifolds, i.e., toric varieties endowed with $T$-invariant holomorphic Poisson structures.