Extensions of Poisson Structures on Singular Hypersurfaces

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 $\mathbb{C}^n$.