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Nonprojective crepant resolutions of quiver varieties

Daniel Kaplan, Travis Schedler

TL;DR

This paper studies nonprojective crepant resolutions of Nakajima quiver varieties, i.e., proper symplectic resolutions of the singularities $\mathcal M_{0}(Q,\alpha)$, by developing two construction paradigms. The first builds local crepant resolutions by gluing parabolic Springer-type models around symplectic leaves, yielding many global resolutions; the second uses good quotients $U//G_{\alpha}$ of $G$-stable open sets $U\subseteq \mu^{-1}(0)$ to obtain resolutions that may not factor through any projective partial resolution. The authors provide numerous explicit examples (including 4- and 6-vertex and some 3-vertex quivers) and prove a general result (Theorem t:stars-git) giving proper crepant resolutions that do not factor through any projective partial resolution. They relate their approach to toric hyperkähler geometry, Cox-ring frameworks, and prior work by Arbo–Proudfoot and Hubbard, expanding the landscape of nonprojective crepant resolutions in the Nakajima setting.

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 hyperkähler, 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.

Nonprojective crepant resolutions of quiver varieties

TL;DR

This paper studies nonprojective crepant resolutions of Nakajima quiver varieties, i.e., proper symplectic resolutions of the singularities , by developing two construction paradigms. The first builds local crepant resolutions by gluing parabolic Springer-type models around symplectic leaves, yielding many global resolutions; the second uses good quotients of -stable open sets to obtain resolutions that may not factor through any projective partial resolution. The authors provide numerous explicit examples (including 4- and 6-vertex and some 3-vertex quivers) and prove a general result (Theorem t:stars-git) giving proper crepant resolutions that do not factor through any projective partial resolution. They relate their approach to toric hyperkähler geometry, Cox-ring frameworks, and prior work by Arbo–Proudfoot and Hubbard, expanding the landscape of nonprojective crepant resolutions in the Nakajima setting.

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 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 hyperkähler, 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.
Paper Structure (30 sections, 23 theorems, 42 equations)

This paper contains 30 sections, 23 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Crepant resolutions of singularities
  3. Constructions using locally projective resolutions
  4. Constructions using good quotients of open subsets
  5. Relation to the work of Arbo--Proudfoot AP16
  6. Relation to the work of Hubbard Hubbard
  7. Framed vs unframed quivers
  8. Preliminary results
  9. Nakajima quiver varieties
  10. Definition
  11. Local neighborhoods
  12. Resolutions of singularities and stratifications
  13. Good quotients
  14. Locally projective constructions
  15. Four-pointed star example
  16. ...and 15 more sections

Key Result

Theorem 2.2

BCS23 If $\alpha$ is a dimension vector with some $\alpha_i = 1$, such that there exists a simple representation in $\mu^{-1}(0)$, then the projective crepant partial resolutions of $\mathcal{M}_{0}(Q, \alpha)$ are precisely the maps $\mathcal{M}_{\theta}(Q, \alpha) \to \mathcal{M}_0(Q,\alpha)$ for

Theorems & Definitions (59)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 2.1: Nakajima
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 2.7
  • ...and 49 more