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
$\bC^2/G$ is the only resolution with trivial canonical bundle and McKay
formulated the theorem as a correspondence between representations of
$\Gamma$ and the configuration of the projective lines in the exceptional
fiber. (This fiber is a chain of $\bP^1$'s such that 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 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.