McKay correspondence for symplectic quotient singularities

Abstract

Let V be a finite-dimensional complex vector space, and G ⊂ SL(V ) a finite subgroup. The quotient variety X = V/G is usually singular. If dim V = 2, then there exists a canonical smooth projective resolution π : Y → X of the singular variety X. This resolution is crepant, which in this case means that the manifold Y has trivial canonical bundle (we recall the precise definitions in Definition 1.2). The fiber π−1(0) ⊂ Y over the singular point 0 ∈ X = V/G is a rational curve, whose components are numbered by the nontrivial conjugacy classes in G. The homology classes of these components freely generate H2(Y,Q). All the other homology groups of Y are trivial, except for H0(Y,Q) ∼= Q. This situation was described by J. McKay in [McK]. It is known as the McKay correspondence. In higher dimensions the picture splits into two parts. One can consider two separate questions. 1. When does the quotient X = V/G admit a smooth crepant resolution Y → X? 2. Assuming there exists a smooth crepant resolution Y → X, what can one say about the homology H (Y,Q)?