Søren Gammelgaard: Quiver varieties and moduli spaces attached to Kleinian singularities

Abstract

We discuss how Nakajima's quiver varieties can be used to understand the Hilbert schemes of points on Kleinian singularities, that is, singular surfaces isomorphic to \(\mathbb{C}^2/G\), for G a finite subgroup of \(\mathrm{SL}_2(\mathbb{C})\).

This approach lets us show that each such Hilbert scheme is irreducible, has symplectic singularities, and has a unique symplectic resolution.

Time permitting, we may touch upon a type of “equivariant Quot scheme” associated to the same singularity, or possibly a generalisation to sheaves on a stacky “compactification” of the same singular surface.

This is joint work with A. Craw, Á. Gyenge, and B. Szendrői.