Workshop on McKay correspondence for symplectic resolutions (following Bezrukavnikov-Kaledin)

(January 25-26, 2008 at University of Amsterdam, together with University Duisburg-Essen)
(H. Esnault, M. Lehn, J. Heinloth)
The aim of the Workshop is to understand the proof, due to Bezrukavnikov and Kaledin (McKay equivalence for symplectic resolutions of quotient singularities, arXiv:math/0401002), of a higher dimensional generalization of the so called McKay correspondence. This proof uses a rather large variety of ideas, so we hope that the proof itself might be as interesting as the result:

Theorem (Bezrukavnikov-Kaledin):

Let V be a finite dimensional, complex, symplectic vector space and G a finite subgroup of Sp(V). Assume that we are given a resolution $p: X \to V/G$ such that the symplectic form on the smooth part of $V/G$ extends to a non-degenerate symplectic form on X. Then the derived category of X is equivalent to the equivariant derived category of V.

In the original version of McKay the above result was considered in the case V=C^2. In this case the minimal resolution of the surface singularity C^2/G is the only resolution with trivial canonical bundle and McKay formulated the theorem as a correspondence between representations of G and the configuration of the projective lines in the exceptional fiber. (This fiber is a chain of $P^1$'s and their intersections are given by one of the Dynkin diagrams without multiple lines - the A,D,E diagrams.)
Bridgeland, King and Reid then proved that one could regard this as an equivalence of derived categories and that a generalization holds in dimension 3. (Here the G-Hilbertscheme is a crepant resolution). This is one precise formulation of the more general idea that the geometry of a crepant resolution should be encoded in the equivariant geometry of V.
The proof of Bezrukavnikov and Kaledin is a surprising combination of ideas, using results on derived categories as discussed in the Intercity-Seminar, quantizations (i.e. non-commutative deformations of the structure sheaves of the varieties occuring in the above theorem) and reduction to positive charactersitic, where these non-commutative algebras turn out to be Azumaya-algebras, closely related to the algebra of differential operators, as discussed in the Forschungsseminar in Essen.
As these ideas are applied in the rather explicit situation of a quotient of a vector space by a finite group, this should be also an opportunity to learn what the above mentioned techniques are about.

Here is the detailed program describing the contents of the talks.

Program

(Korteweg-de Vries Institute for Mathematics Plantage Muidergracht 24, 1018 TV Amsterdam)

Friday, January 25, 2008

12:00-13:00 lunch (we can just go to the Mensa near the institute, where you can get sandwiches, soup and salad)
13:00-14:15 H. Esnault: The case of dimension 2 and almost exceptional objects
coffee/tea
14:45-16:00 S. Kukulies: A noncommutative version - quantizations and Azumaya algebras
coffee/tea
16:30-17:45 R. Brussee:The quantization in case dim=2
Dinner

Saturady, January 26, 2008

9:15-10:30 J. Heinloth: The new proof in dimension 2 - untwisting
coffe/tea
10:45-12:00 M. Lehn: The quantization in arbitrary dimension

For more information you can send me an email: (my-last-name)@science.uva.nl.
[back to my homepage] 

Last modified: 14.01.2008     Verantwortlicher: Jochen Heinloth (email: use @science.uva.nl with heinloth)