Stacky Resolutions of Kleinian Singularities and Nakajima Quiver Varieties

Abstract

We construct a class of noncommutative crepant resolutions of any Kleinian singularity as noncommutative sheaves of algebras over its crepant partial resolutions. We argue that such resolutions are Morita equivalent to the canonical orbifold resolutions of the partial resolutions. Further, we study Hilbert schemes of points on both the crepant partial and noncommutative resolutions and show that they are Nakajima quiver varieties. Finally, we describe the nef and movable cones of these Hilbert schemes of points using tautological bundles on the crepant partial resolutions.

40 pages. v2 restructured and expanded: Added results on the crepant partial resolutions parallel to those on the noncommutative/stacky resolutions contained in v1 (sections 1.7, 4); Added a description of the nef and movable cones of the Hilbert schemes of points (section 6); Added lemmas, propositions, explanations, references (sections 1, 2, 3, 5)