Noncommutative projective partial resolutions and quiver varieties

Abstract

Let $\hat{I}“\in {\mathrm{SL}}_2(\mathbb{C})$ be a finite subgroup. We introduce a class of projective noncommutative surfaces ${\mathbb{P}}_I^2$, indexed by a set of irreducible $\hat{I}“$-representations. Extending the action of $\hat{I}“$ from ${\mathbb{C}}^2$ to ${\mathbb{P}}^2$, we show that these surfaces generalise both $[{\mathbb{P}}^2/\hat{I}“]$ and ${\mathbb{P}}^2/\hat{I}“$. We prove that isomorphism classes of framed torsion-free sheaves on any ${\mathbb{P}}_I^2$ 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.

31 pages. Comments are welcome v2: corrected several small mistakes and typos v3: corrected homological algebra errors, cleaned up proofs