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Poisson (co)homology and isolated singularities. (English) Zbl 1113.17009
Let \(\mathbb{F}\) be a field of characteristic \(0\) and \(\mathcal{A}=\mathbb{F}[x,y,z]\). Given any \(\varphi\in \mathcal{A}\), the relations \(\{x,y\}_{\varphi}=\frac{\partial \varphi}{\partial z}\), \(\{y,z\}_{\varphi}=\frac{\partial \varphi}{\partial x}\), \(\{z,x\}_{\varphi}=\frac{\partial \varphi}{\partial y}\) define a Poisson bracket on \(\mathcal{A}\), which admits \(\varphi\) as a Casimir function. Therefore, this bracket induces Poisson structures both on the affine three space \(F^{3}\) and the surface \(\{\varphi=0\}\subset F^{3}\). Suppose that \(\varphi\) is a weighted homogeneous polynomial such that the surface \(\{\varphi=0\}\) has an isolated singularity at the origin. The author computes the Poisson cohomology and homology modules of the Poisson structures on \(F^{3}\) and \(\{\varphi=0\}\) in this case. The paper also contains clear explanations of each of the concepts mentioned.

17B63 Poisson algebras
14F99 (Co)homology theory in algebraic geometry
17B56 Cohomology of Lie (super)algebras
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