Quot schemes for Kleinian orbifolds

TL;DR

This paper establishes a connection between orbifold Quot schemes for Kleinian orbifolds and Nakajima quiver varieties, providing new geometric insights and generalizing previous results without relying on ADE classification.

Contribution

It identifies moduli spaces of cornered quiver algebras with orbifold Quot schemes and describes their structure as Nakajima quiver varieties at specific stability parameters.

Findings

Orbifold Quot schemes are irreducible and normal.

These schemes admit symplectic resolutions.

The results generalize previous work on Hilbert schemes without ADE classification.

Abstract

For a finite subgroup $\Gamma\subset {\mathrm{SL}}(2,\mathbb{C})$, we identify fine moduli spaces of certain cornered quiver algebras, defined in earlier work, with orbifold Quot schemes for the Kleinian orbifold $[\mathbb{C}^2/\Gamma]$. We also describe the reduced schemes underlying these Quot schemes as Nakajima quiver varieties for the framed McKay quiver of $\Gamma$, taken at specific non-generic stability parameters. These schemes are therefore irreducible, normal and admit symplectic resolutions. Our results generalise our previous work on the Hilbert scheme of points on $\mathbb{C}^2/\Gamma$; we present arguments that completely bypass the ADE classification.