Formal Poisson (co)homology of the Lefschetz singularity

We compute the formal Poisson cohomology groups of a real Poisson structure π on C² associated to the Lefschetz singularity (z₁, z₂)↦ z₁²+z₂². In particular we correct an erroneous computation in the literature. The definition of π depends on a choice of volume form. Using the main result we formally classify all Poisson structure arising from different choices of volume forms.