Poisson cohomology of plane Poisson structures with isolated singularities revisited

TL;DR

This paper revisits the Poisson cohomology of plane Poisson structures with isolated singularities, revealing a GAGA phenomenon and providing explicit descriptions for simple singularities.

Contribution

It explicitly describes the Gerstenhaber algebra structure of Poisson cohomology for structures with isolated singularities, extending previous work.

Findings

Identification of a GAGA type phenomenon in Poisson cohomology

Explicit generators and relations for simple singularities

Extension of previous cohomology structure results

Abstract

Continuing a work of Ph.~Monnier, we determine the Gerstenhaber algebra structure over the Poisson cohomology groups for a large class of Poisson structures with isolated singularities over the plane. It reveals that there exists a GAGA type phenomenon. We give an explicit description via generators and relations for the case of simple singularities.