Nonprojective crepant resolutions of quiver varieties

TL;DR

This paper constructs many examples of proper, nonprojective crepant resolutions for Nakajima quiver varieties, expanding the understanding of their geometric structures through novel techniques involving quotients and hyperkähler geometry.

Contribution

It introduces new methods for constructing nonprojective crepant resolutions of quiver varieties, including quotients of unstable loci, and provides explicit examples in various dimensions.

Findings

Constructed large classes of nonprojective crepant resolutions

Developed techniques involving quotients of unstable loci

Connected hyperkähler resolutions to toric geometry

Abstract

In this paper, we construct a large class of examples of proper, nonprojective crepant resolutions of singularities for Nakajima quiver varieties. These include four and six dimensional examples and examples with $Q$ containing only three vertices. There are two main techniques: by taking a locally projective resolution of a projective partial resolution as in our previous work arXiv:2311.07539, and more generally by taking quotients of open subsets of representation space which are not stable loci, related to Arzhantsev--Derental--Hausen--Laface's construction in the setting of Cox rings. By the latter method we exhibit a proper crepant resolution that does not factor through a projective partial resolution. Most of our quiver settings involve one-dimensional vector spaces, hence the resolutions are toric hyperk\"ahler, which were studied from a different point of view in Arbo and Proudfoot arXiv:1511.09138. This builds on the classification of projective crepant resolutions of a large class of quiver varieties in arXiv:2212.09623 and the classification of proper crepant resolutions for the hyperpolygon quiver varieties in arXiv:2406.04117.