University of Toronto's Symplectic Geometry Seminar

March 29, 2004, 2:10 - 3 PM

Andre Henriques


Orbifolds as stacks versus orbifolds as non-commutative spaces, and the resolution of ADE surface singularities

Abstract: The ADE surface singularities are the quotients of C^2 by finite subgroups of SU2. They are very natural examples of complex orbifolds. Their minimal resolutions Y->X carry the information of the corresponding Dynkin diagram. To make sense of the map Y->X, we usually view X and Y as varieties. But in that context, X is clearly singular, which is contrary to the spirit of orbifolds. Somehow, we should endow X with some structure that makes it behave as if it were a smooth variety. If we decide to view X as a smooth stack, then we notice that the map Y->X doesn't exist any more, which is annoying. However, if we view X as a non-commutative space, then the map X->Y does exist.