Partial Resolutions of Affine Symplectic Singularities

Abstract: We explore the relationship between the Poisson deformation theory, birational geometry, and Springer theory of partial resolutions of affine symplectic singularities. Let $\rho: X' \rightarrow X$ be a crepant partial resolution of a conical affine symplectic singularity $X$. We show that the Poisson deformation functor of $X'$ is prorepresentable and unobstructed. Additionally, we define a version of the Namikawa Weyl group for these crepant partial resolutions. In particular, the Namikawa Weyl group of $X'$ is a parabolic subgroup of the Namikawa Weyl group of $X$ that is determined by the birational geometry of $X'$. If $\pi: Y \rightarrow X$ is a $\mathbb{Q}$-factorial terminalization of $X$ that covers $X'$, we show there is a natural functor from Poisson deformations of $Y$ to those of $X'$. Building on work of Namikawa, we show that this morphism is a Galois covering and the Galois group is the Namikawa Weyl group of $X'$. Finally, we put these partial resolutions and their universal deformations into the context of recent work of McGerty and Nevins, obtaining some preliminary results concerning their Springer theory. In particular, if the universal deformation of $X'$ is rationally smooth, we compute the cohomology of the fibers of $\rho$ in terms of the cohomology of the fibers of $\pi$ and the Namikawa Weyl group of $X'$.