Poisson (co)homology and isolated singularities

To each polynomial $\check{\in }\text{F}[x,y,z]$ is associated a Poisson structure on ${\text{F}}^3$, a surface and a Poisson structure on this surface. When \v is weight homogeneous with an isolated singularity, we determine the Poisson cohomology and homology of the two Poisson varieties obtained.