Quiver Varieties and Framed Sheaves on Compactified Kleinian Singularities

Consider a Kleinian singularity [Formula: see text], where [Formula: see text] is a finite subgroup of SL 2 ([Formula: see text]). 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.