On Toric Poisson Structures of Type $(1,1)$ and their Cohomology

TL;DR

This paper classifies certain Poisson structures on complex toric manifolds, explores their algebraic properties, and computes their low-degree Poisson cohomology, providing insights into their geometric and algebraic features.

Contribution

It provides a classification of real Poisson structures of type (1,1) on complex toric manifolds and computes their low-degree cohomology groups under specific conditions.

Findings

Poisson structures are algebraic and quadratic in coordinate charts.

Computed H^0 and H^1 cohomology groups for non-degenerate structures.

Numerical analysis of higher cohomology groups for specific cases.

Abstract

We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in each of the distinguished holomorphic coordinate charts determined by the open cones of the associated simplicial fan. As an approximation to the smooth cohomology problem in each ${\mathbb C}^n$ chart, we consider the Poisson differential on the complex of polynomial multi-vector fields. For the algebraic problem, we compute $H^0$ and $H^1$ under the assumption that the Poisson structure is generically non-degenerate. The paper concludes with numerical investigations of the higher degree cohomology groups of $({\mathbb C}^2,\pi_B)$ for various $B$.