Algebra Seminar

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 CC²/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.

[This is the abstract for the talk cited above.]
M. Reid, La correspondance de McKay, Séminaire Bourbaki, 52ème année, novembre 1999, no. 867, to appear in Astérisque 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 Cⁿ 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_M=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.