University of Sydney
School of Mathematics and Statistics
Miles Reid
University of Warwick
McKay correspondence, theme and free variations
Friday 27th July, 12-1pm,
Carslaw 375.
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
KM=0 and G a finite automorphism group of
M acting trivially on the canonical class
KM; 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 KM=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.