Punctual Hilbert schemes for Kleinian singularities as quiver varieties

TL;DR

This paper constructs Hilbert schemes of points on Kleinian singularities as Nakajima quiver varieties, proving their irreducibility, normality, and existence of a unique symplectic resolution, and introduces new algebras related to these structures.

Contribution

It provides a novel realization of Hilbert schemes on Kleinian singularities as quiver varieties using non-generic stability parameters, and introduces cornered algebras with moduli spaces isomorphic to these quiver varieties.

Findings

Hilbert schemes are irreducible and normal.

Hilbert schemes admit a unique symplectic resolution.

New classes of algebras are related to quiver varieties.

Abstract

For a finite subgroup $\Gamma\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/\Gamma$ as a Nakajima quiver variety for the framed McKay quiver of $\Gamma$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, and we show that fine moduli spaces of cyclic modules over these new algebras are isomorphic to quiver varieties for the framed McKay quiver and certain non-generic choices of stability parameter.