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

In this paper, we show that for a given degenerate bivector $\pi =y^n{\mathrm{\partial }}_x\wedge {\mathrm{\partial }}_y$ 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 dégrée. 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.