Noncommutative projective partial resolutions and quiver varieties

TL;DR

This paper introduces a new class of noncommutative projective surfaces that generalize classical quotients and establishes a geometric link between sheaves on these surfaces and Nakajima quiver varieties, enriching the understanding of their structure.

Contribution

It defines the surfaces $ ext{P}^2_I$ associated with finite subgroups of SL(2,C), extending group actions and connecting sheaf moduli to Nakajima quiver varieties, thus broadening the geometric framework.

Findings

Classified isomorphism classes of sheaves with points of Nakajima quiver varieties.

Established geometric interpretations of certain Nakajima quiver varieties.

Generalized previous results on quiver varieties using noncommutative geometry.

Abstract

Let $\Gamma\in \mathrm{SL}_2(\mathbb{C})$ be a finite subgroup. We introduce a class of projective noncommutative surfaces $\mathbb{P}^2_I$, indexed by a set of irreducible $\Gamma$-representations. Extending the action of $\Gamma$ from $\mathbb{C}^2$ to $\mathbb{P}^2$, we show that these surfaces generalise both $[\mathbb{P}^2/\Gamma]$ and $\mathbb{P}^2/\Gamma$. We prove that isomorphism classes of framed torsion-free sheaves on any $\mathbb{P}^2_I$ carry a canonical bijection to the closed points of appropriate Nakajima quiver varieties. In particular, we provide geometric interpretations for a class of Nakajima quiver varieties using noncommutative geometry. Our results partially generalise several previous results on such quiver varieties.