Manchester Geometry Seminar 2013/2014

Thursday 20 March 2014. The Frank Adams Room (Room 1.212), the Alan Turing Building. 4.15pm

Power Structure on the Grothendieck Ring of Varieties and Motivic Formula for the Calogero--Moser Spaces on Curves Oleg Chalykh (University of Leeds)

oleg@amsta.leeds.ac.uk

I will explain, following Gusein-Zade, Luengo and Melle-Hernandez, how to define a power structure on the Grothendieck ring of complex algebraic varieties. Gusein-Zade et al. used this to obtain a nice motivic formula for the punctual Hilbert scheme on surfaces. Recall that for a smooth complex algebraic surface $X$, the Hilbert scheme $\mathbf{Hilb}_n(X)$ of $n$ points on $X$ is a certain desingularization of $\mathbf{S}^n(X)$. A similar approach can be applied to the Calogero--Moser spaces associated to a smooth algebraic curve $C$. These spaces arise in the context of Cherednik algebras and they give a desingularization of $\mathbf{S}^n(T^*C)$, different from $\mathbf{Hilb}_n(T^*C)$. As an application, the Euler characteristic and Deligne-Hodge polynomial of these spaces can be computed.