QUIVER VARIETIES AND HILBERT SCHEMES

In this note we give an explicit geometric description of some of the Nakajima's quiver varieties. More precisely, if X = C-2, Gamma subset of SL(C-2) is a finite subgroup, and X-Gamma is a minimal resolution of X/Gamma, we show that X-Gamma[n] (the Gamma-equivariant Hilbert scheme of X), and X-Gamma([n]) (the Hilbert scheme of X-Gamma) are quiver varieties for the affine Dynkin graph corresponding to via the McKay correspondence with the same dimension vectors but different parameters zeta (for earlier results in this direction see works by M. Haiman, M. Varagnolo and E. Vasserot, and W. Wang). In particular, it follows that the varieties X-Gamma[n] and X-Gamma([n]) are diffeomorphic. Computing their cohomology (in the case = Z/dZ) via the fixed points of a (C* x C*)-action we deduce the following combinatorial identity: the number UCY (n, d) of Young diagrams consisting of nd boxes and uniformly colored in d colors equals the number UCY (n, d) of collections of d Young diagrams with the total number of boxes equal to n.