Quiver Varieties and Framed Sheaves on Compactified Kleinian Singularities

Consider a Kleinian singularity ${\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a finite subgroup of $S{L}_2(\mathbb{C})$. In this paper, we introduce a natural stack compactifying the singularity by adding a smooth stacky divisor, and we show that sets of framed sheaves on this stack satisfying certain additional criteria are closely related to a class of Nakajima quiver varieties. This partially extends our previous work on punctual Hilbert schemes of Kleinian singularities.