The McKay Correspondence

Alastair King

For a finite subgroup G of SL(n, C) one may consider the ordinary (commutative) geometry of crepant resolutions of the quotient singularity C^{n}/G and also the (mildy non-commutative) G-equivariant geometry of C^{n} itself. The McKay correspondence — first observed for C^{2}, but now much studied for C^{3} — describes the relationship between these two.

This mini-course will survey some of the history of the correspondence from Gonzalez-Sprinberg and Verdier’s K-theoretic interpretation of McKay’s original observation, to the more modern interpretation as an equivalence of derived categories of sheaves on the resolution and of equivariant sheaves on C^{n}.

The course will also describe some more recent progress, including Craw & Ishii’s result that all projective crepant resolutions of abelian quotients of C^{3} can be realised as moduli of equivariant sheaves and Logvinenko’s theory of G-equivariant divisors.