The Mckay Correspondence for Finite Subgroups of Sl(3,c)

Let G ⊂ SL(n, C) be a finite subgroup and φ:Y → X = C/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 KY = 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.