The McKay correspondence for finite subgroups of SL (3, C)

Abstract

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).

Mass Formulas for Local Galois Representations and Quotient Singularities. I: A Comparison of Counting Functions

“…The weight w τ (g) defined above in the tame case is essentially the same thing as what is called age [IR96] or fermion number shift [Zas93].…”

Group representations and the Euler characteristic of elliptically fibered Calabi–Yau threefolds

“…Much recent work has been done by looking at the quotient of C 3 by a finite subgroup of SL(3, C) (see for example [20,19,30]). In this paper we consider another natural generalization of the above set up.…”

Hilbert schemes of $G$-orbits in dimension three

“…There is a unique junior conjugacy class with all eigenvalues of ρ 3 1 distinct from one. Hence by [IR96] …”

“…We note thatρ k ρ k for any k and ρ = ρ 3 1 ρ ∨ . We also note that ρ 3 1 (g) has an eigenvalue one for g belonging to any junior conjugacy class, whence by [IR96], there is no compact irreducible divisor in the fibre of the Hilbert-Chow morphism over the origin. See Corollary 3.6.…”