Partial Resolutions of Affine Symplectic Singularities
Alberto San Miguel Malaney
TL;DR
The paper advances the theory of partial resolutions of conical affine symplectic singularities by establishing a Poisson-theoretic and birational framework that parallels Springer theory. It defines a Namikawa Weyl group for crepant partial resolutions ρ: X'→X and proves that the Poisson deformation functor $PD_{X'}$ is prorepresentable and unobstructed, with a universal conic deformation that fits into a diagram where $PD_{X'}$ is the $W_{X'}$-quotient of the universal deformation of a covering $\,Q$-factorial terminalization. It also shows that pushforward deformations along a covering chain $Y o X' o X$ are Galois with group $W_{X'}$, and connects these deformations to recent Springer-theoretic work, detailing when fiber cohomology satisfies $H^*(\rho^{-1}(x)) \cong H^*(\pi^{-1}(x))^{W_{X'}}$. The results unify deformation theory, Mori theory, and symplectic Springer theory, and provide concrete computations of fiber cohomology under rational smoothness of the universal deformation, highlighting the deep interaction between geometry and representation theory in this setting.
Abstract
We explore the relationship between the Poisson deformation theory, birational geometry, and Springer theory of partial resolutions of affine symplectic singularities. Let $ρ: X' \rightarrow X$ 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 $π: Y \rightarrow X$ 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 $ρ$ in terms of the cohomology of the fibers of $π$ and the Namikawa Weyl group of $X'$.