$DG$-models of projective modules and Nakajima quiver varieties

Abstract

Associated to each finite subgroup $\Gamma$ of ${\tt SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O^\tau(\Gamma)$, which is a deformation of the coordinate ring of the Kleinian singularity $\mathbb{C}^2/\Gamma$. We study finitely generated projective modules over these algebras. Our main result is a bijective correspondence between the set of isomorphism classes of rank one projective modules over $O^\tau$ and a certain class of quiver varieties associated to $\Gamma$s. We show that this bijection is naturally equivariant under the action of a “large” Dixmier-type automorphism group $G$. Our construction leads to a completely explicit description of ideals of the algebras $O^\tau$ .