Abstract: |
For a finite subgroup G of SL(2,C), a well known construction of Kronheimer realises the minimal resolution S of the Kleinian singularity C^2/G as a quiver variety, and implies that the ample cone of S is a Weyl chamber in the root system of type ADE associated to G by the McKay correspondence. For any n>1, there is a natural generalisation to dimension 2n, namely, the Hilbert scheme X=Hilb^[n](S) of n points on S, which is itself a resolution of a symplectic quotient singularity C^{2n}/G_n; here, G_n is the wreath product of G with S_n. In dimension greater than two, minimal resolutions are not unique, so one expects the situation to be much more complicated. While this is the case, I'll explain why every projective, crepant resolution of C^{2n}/G_n is a quiver variety and, in addition, why the movable cone of X can be described in terms of an extended Catalan hyperplane arrangement of the root system associated to G. The Namikawa Weyl group arises naturally in this context, and it plays a key role. This is joint work with Gwyn Bellamy. |