The McKay correspondence is a principle that relates the geometry of a
resolution of singularities of a quotient variety M/G and the equivariant
geometry of the group action. The classic case is McKay's identification of
the cohomology of the resolution of the Klein quotient singularities CC^2/G
with the representation theory of G. This principle and its applications to
different geometric categories are described in my Bourbaki talk cited
below (much of which can be read as a colloquial presentation).
The talk will head in the direction of some recent developments, including
interpretation of crepant resolutions as moduli spaces of Artinian G-modules
on M, and flops between them as variation of GIT quotient (work of Alastair
Craw, Akira Ishii, and Alastair King). The title of the talk includes the
horrible little pun: free variation = unobstructed deformation.
M. Reid, La correspondance de McKay, S\'eminaire Bourbaki, 52\`eme ann\'ee,
novembre 1999, no. 867, to appear in Ast\'erisque 2001, preprint
math/9911165, 20 pp.
Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite
automorphism group of M acting trivially on the canonical class K_M; for
example, a subgroup G of SL(n,C) acting on C^n in the obvious way. We aim
to study the quotient variety X=M/G and its resolutions Y -> X (especially
under the assumption that Y has K_Y=0) in terms of G-equivariant geometry
of M. At present we know 4 or 5 quite different methods of doing this,
taken from string theory, algebraic geometry, motives, moduli, derived
categories, etc. For G in SL(n,C) with n=2 or 3, we obtain several methods
of cobbling together a basis of the homology of Y consisting of algebraic
cycles in one-to-one correspondence with the conjugacy classes or the
irreducible representations of G.
Created: 4/25/01 Updated: 4/25/01