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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.

MSC:
 17B63 Poisson algebras 14F99 Homology and cohomology theory (algebraic geometry) 17B56 Cohomology of Lie (super)algebras
Full Text:
References:
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