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Symplectic resolutions of quiver varieties

Gwyn Bellamy, Travis Schedler

Abstract

In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically $θ$-polystable points, generalizing a result of Le Bruyn; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

Symplectic resolutions of quiver varieties

Abstract

In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically -polystable points, generalizing a result of Le Bruyn; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

Paper Structure

This paper contains 37 sections, 64 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Symplectic resolutions
  3. Symplectic leaves and the étale local structure
  4. Factoriality of quiver varieties
  5. Smooth versus canonically polystable points
  6. Namikawa's Weyl group
  7. Character varieties
  8. Applications
  9. Other related work
  10. Acknowledgments
  11. Notation and proof of the main results
  12. Quiver varieties
  13. Notation
  14. Root systems
  15. The canonical decomposition
  16. ...and 22 more sections

Key Result

Theorem 1.2

The variety $\mathfrak{M}_{{\lambda}} ({\alpha},{\theta})$ is an irreducible symplectic singularity.

Theorems & Definitions (123)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 113 more