Let G ⊂ SL(n, C) be a finite subgroup and ϕ: Y → X = C n /G any resolution of singularities of the quotient space. We prove that crepant exceptional prime divisors of Y correspond one-to-one with "junior" conjugacy classes of G. When n = 2 this is a version of the McKay correspondence (with irreducible representations of G replaced by conjugacy classes). In the case n = 3, a resolution with K Y = 0 is known to exist by work of Roan and others; we prove the existence of a basis of H * (Y, Q) by algebraic cycles in one-to-one correspondence with conjugacy classes of G. Our treatment leaves lots of open problems. Contents1. Statement of the results 2. Proofs 3. Examples 4. Discussion References Statement of the resultsLet G ⊂ SL(n, C) be a finite subgroup and X = C n /G the quotient space, an affine variety with K X = 0. A crepant resolution f : Y → X is a resolution of singularities such that K Y = f * K X = 0. A crepant resolution does not necessarily exist in dimension ≥ 4 (see 4.5); it is known to exist in dimension 2 (classical, Du Val), and in dimension 3 by work of a number of people (see for example [Markushevich], [Ito1-3] and [Roan]), but the proofs are computational rather than conceptual.This paper contributes some very easy remarks to the following question raised by [Dixon-Harvey-Vafa-Witten], worked out by [Hirzebruch-Höfer], and now famous among algebraic geometers as the "Physicists' Euler number conjecture". See for example [Roan] for the background.Conjecture 1.1. G ⊂ SL(n, C) is a finite subgroup, X = C n /G the quotient space and f : Y → X a crepant resolution. Then there exists a basis of H * (Y, Q) consisting of algebraic cycles in one-to-one correspondence with conjugacy classes of G.It is an elementary fact that Y has no odd-dimensional cohomology, and that H 2i (Y, Q) is spanned by algebraic cycles (see 4.1).

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