Weighted Poisson polynomial rings in dimension three

Abstract

We discuss Poisson structures on a weighted polynomial algebra $A:=\mathbb{k}[x,y,z]$ defined by a homogeneous element $\Omega \in A$, called a potential. We start with classifying potentials $\Omega$ of degree $\mathrm{deg}(x)+\mathrm{deg}(y)+\mathrm{deg}(z)$ with any positive weight $(\mathrm{deg}(x),\mathrm{deg}(y),\mathrm{deg}(z))$ and list all with isolated singularity. Based on the classification, we study the rigidity of $A$ in terms of graded twistings and classify Poisson fraction fields of $A/(\Omega )$ for irreducible potentials. Using Poisson valuations, we characterize the Poisson automorphism group of $A$ when $\Omega$ has an isolated singularity extending a nice result of Makar-Limanov–Turusbekova–Umirbaev. Finally, Poisson cohomology groups are computed for new classes of Poisson polynomial algebras.