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, Kleinian singularities, and affine grassmannian slices. Namikawa proved in 2013 that a symplectic resolution Y of a conical affine symplectic singularity X is in fact a relative Mori Dream Space over X. This result allows us to give a combinatorial description of the crepant partial resolutions of X, showing they are in bijection with faces of the movable cone Mov(Y/X). 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 crepant partial resolutions. We will also use this Namikawa Weyl group to classify the Poisson deformations of the partial resolutions. Next, we will 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, I will briefly discuss how we can use partial resolutions of symplectic singularities to construct and study symplectic resolutions of the normalizations of symplectic leaf closures.
Algebraic Geometry Seminar
Monday, March 17, 2025 - 3:30pm
Alberto San Miguel Malaney
UGA
Other Events on This Day
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Proof Complexity and “quantitative” model theory (via trees, polynomials, and games)
Logic and Computation Seminar
4:30pm