Symplectic resolutions of quiver varieties

TL;DR

This paper studies Nakajima quiver varieties as symplectic singularities, classifies those with symplectic resolutions, and explores their local structure, revealing that not all resolutions arise from GIT variation.

Contribution

It provides a complete classification of symplectic resolutions of Nakajima quiver varieties and analyzes their local and global symplectic structures.

Findings

All Nakajima quiver varieties are symplectic singularities.

The paper classifies which quiver varieties admit symplectic resolutions.

Not all symplectic resolutions are obtained via GIT variation.

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 $\theta$-polystable points, generalizing a result of Le Bruyn; we study their \'etale 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.