Extensions of Poisson Structures on Singular Hypersurfaces

TL;DR

This paper studies how Poisson structures on singular hypersurfaces can be extended to ambient spaces, providing a characterization using algebraic tools and proving existence results for certain cases.

Contribution

It offers a new algebraic characterization of Poisson structure extensions on singular hypersurfaces and proves the existence of extensions for singular surfaces.

Findings

Characterization of Poisson structure extensions via the Koszul complex

Existence of extensions for singular surfaces

Framework applicable to affine Poisson varieties with isolated singularities

Abstract

Fix a codimension-1 affine Poisson variety $(X,\pi_X)$ in $\mathbb{C}^n$ with an isolated singularity at the origin. We characterize possible extensions of $\pi_X$ to $\mathbb{C}^n$ using the Koszul complex of the Jacobian ideal of $X$. In the particular case of a singular surface, we show that there always exists an extension of $\pi_X$ to $\mathbb{C}^n$.