Birational geometry of symplectic quotient singularities

Alastair Craw (Bath)

Monday 29th October, 2018 16:00-17:00 Maths 311B


For a finite subgroup G of SL(2,C), a well known result of Kronheimer constructs the minimal resolution S of the Kleinian singularity C^2/G as a quiver variety. 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. I'll describe that this is indeed the case (!), but I'll also show that the situation can be understood completely; in fact, every projective, crepant resolution of C^{2n}/G_n is a quiver variety. No prior knowledge of quiver varieties or birational geometry will be assumed.

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