Seminars in Symplectic Geometry

We will continue to discuss partial resolutions of conical affine symplectic singularities, particularly their deformation theory and Springer theory. First we will explain the construction of the universal deformations of symplectic singularities and their partial resolutions, generalizing the Grothendieck-Springer resolution. Then we will use these universal deformations to study the Springer theory of symplectic singularities and their partial resolutions, using recent work of McGerty and Nevins. In particular, we will compute the cohomology of the fibres of the partial resolutions under suitable conditions, generalizing a result of Borho and MacPherson for the nilpotent cone. Finally, we will use partial resolutions to construct and study symplectic resolutions of symplectic leaf closures, generalizing the Springer maps from cotangent bundles of partial flag varieties to nilpotent orbit closures.

Symplectic singularities are a generalization of symplectic manifolds that have a symplectic form on the smooth locus but allow for certain well-behaved singularities. They have a strong relationship to representation theory and include nilpotent cones of semisimple Lie algebras, quiver varieties, affine Grassmannian slices, and Kleinian singularities. There is a combinatorial description for partial resolutions of conical affine symplectic singularities, stemming from Namikawa's 2013 result that a symplectic resolution is also a relative Mori Dream Space. In this talk we will explore these partial resolutions in more detail, exploring their birational geometry, deformation theory, and Springer theory. In particular, we will review the definition of the Namikawa Weyl group for conical affine symplectic singularities and use birational geometry to define a generalization for their partial resolutions. We will also use this Namikawa Weyl group to classify the Poisson deformations of the partial resolutions. We will then describe how these partial resolutions fit into the framework of Springer Theory for symplectic singularities, following Kevin McGerty and Tom Nevins' recent paper, Springer Theory for Symplectic Galois Groups. Finally, we will discuss some ongoing research that stems from these ideas, inspired by parabolic induction and restriction.