### 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 *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.

[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 *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*_{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*.