Quiver varieties and framed sheaves on compactified Kleinian singularities

Consider a Kleinian singularity C-2/Gamma, where Gamma is a finite subgroup of SL2(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.