Quiver Varieties and Hilbert Schemes

Abstract

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, we show that the $\Gamma$-equivariant Hilbert scheme $X^{\Gamma[n]}$ and the Hilbert scheme $X_\Gamma^{[n]}$ (where $X=\C^2$, $\Gamma\subset SL(\C^2)$ is a finite subgroup, and $X_\Gamma$ is a minimal resolution of $X/\Gamma$) are quiver varieties for the affine Dynkin graph, corresponding to $\Gamma$ via the McKay correspondence, the same dimension vectors, but different parameters $\zeta$ (for earlier results in this direction see [4, 12, 13]). In particular, it follows that the varieties $X^{\Gamma[n]}$ and $X_\Gamma^{[n]}$ are diffeomorphic. Computing their cohomology (in the case $\Gamma=\Z/d\Z$) via the fixed points of $(\C^*\times\C^*)$-action we deduce the following combinatorial identity: the number $UCY(n,d)$ of uniformly coloured in d colours Young diagrams consisting of nd boxes coincides with the number $CY(n,d)$ of collections of d Young diagrams with the total number of boxes equal to n.