Symplectic resolutions of quiver varieties

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.

Automated summary

In this article, Nakajima quiver varieties are considered from the perspective of symplectic algebraic geometry. The study proves that they are all symplectic singularities according to Beauville and completely classifies which ones admit symplectic resolutions. Additionally, it is shown that the smooth locus aligns with the locus of canonically theta-polystable points, expanding on a previous result by Le Bruyn. The study also examines the etale local structure of these varieties and identifies their symplectic leaves, with an interesting finding that not all symplectic resolutions of quiver varieties seem to originate from variation of GIT.