Poisson deformations of affine symplectic varieties, II

Abstract

This article is a continuation of previous work, which has the same title. Let $Y$ be an affine symplectic variety with a ${\mathbf{C}}^{\ast }$-action with positive weights, and let $\pi :X\to Y$ be its crepant resolution. Then $\pi$ induces a natural map $PDef\left(X\right)\to PDef\left(Y\right)$ of Kuranishi spaces for the Poisson deformations of $X$ and $Y$. In Part I, we proved that $PDef\left(X\right)$ and $PDef\left(Y\right)$ are both nonsingular, and this map is a finite surjective map. In this article (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation for this article. As an application, we construct explicitly the universal Poisson deformation of the normalization $\tilde{O}$ of a nilpotent orbit closure $\tilde{O}$ in a complex simple Lie algebra when $\tilde{O}$ has a crepant resolution.