On logarithmic Poisson cohomology of a degenerate Poisson bivector in affine plane

TL;DR

This paper compares classical and logarithmic Poisson cohomology groups for a specific degenerate bivector in the affine plane, showing they are isomorphic in every degree.

Contribution

It explicitly computes the logarithmic Hamiltonian operator and cochain complex for a degenerate Poisson bivector, establishing their cohomological equivalence.

Findings

Classical and logarithmic Poisson cohomology groups are isomorphic for the given bivector.

The logarithmic Hamiltonian operator is explicitly determined.

The logarithmic Poisson cochain complex is characterized for the bivector.

Abstract

In this paper, we show that for a given degenerate bivector $\pi= y^n\partial_x \wedge \partial_y$ with $n>1$, the classical Poisson cohomology group and the logarithmic Poisson cohomology group along the ideal $\mathcal{I}=y^n\mathbb{F}[x,y] $ are isomorphics in every d\'egr\'ee. This result follows from determination of the logarithmic Hamiltonian operator and the logarithmic Poisson cochain complexe in order to compute the cohomological invariants associated to $\pi$. $\mathbb{F}$ is the field of characteristic 0.