Algebraic Geometry Seminar 2006

Title: On the motive of a Shimura variety
Speaker: Arvind Nair (Tata Institute for Fundamental Research, Mumbai, Maharashtra, India)
Date and Time: Monday, April 24 2006, 01:30 PM - 02:30 PM
Location: West Mall Annex Room 110 (Coffee at 1st floor PIMS lounge), PIMS-UBC
Abstract: Shimura varieties are of interest as a source of motives/Galois representations about which much can be said using the methods of representation theory. For various reasons one would like to have a Grothendieck motive for such a variety. If the Shimura variety is projective this is immediate. If it is not projective (as in most classical examples, e.g. the moduli space of principally polarized abelian varieties), the desired Grothendieck motive should realize to the intersection cohomology of the minimal compactification of Baily and Borel. I'll show that if the Shimura variety is related to abelian varieties, then a motive can be constructed which realizes to the subspace of intersection cohomology satisfying the generalized Ramanujan conjecture at (any) one finite prime. (This is the 'essential' part according to standard conjectures in representation theory.) In the case when the Shimura variety is a modular curve this specializes to the Eichler-Shimura-Deligne-Scholl motive for classical modular forms.

Title: The Hodge conjecture for the primitive cohomology of theta divisors
Speaker: Elham Izadi (University of Georgia in Athens, Georgia, USA)
Date and Time: Monday, April 24 2006, 03:00 PM - 04:00 PM
Location: West Mall Annex Room 110 (Coffee at 1st floor PIMS lounge), PIMS-UBC
Abstract: I will first discuss the meaning of the Hodge conjecture in general and then specialize to abelian varieties. The primitive cohomology of the theta divisor of an abelian variety gives a special Hodge structure for which one can ask whether the Hodge conjecture is true. Using Prym-embedded curves, this question was answered in the affirmative by myself and van Straten for abelian fourfolds. In this talk which is about joint work with Csilla Tamas, I will discuss the case of abelian fivefolds and show in particular that Prym-embedded curves do NOT solve the Hodge conjecture. I will, however, introduce a different family of curves which is very likely to give an answer to the Hodge conjecture.

Title: On the motive of the stack of bundles
Speaker: Kai Behrend (University of British Columbia)
Date and Time: Monday, April 24 2006, 04:30 PM - 05:30 PM
Location: West Mall Annex Room 110 (Coffee at 1st floor PIMS lounge), PIMS-UBC
Abstract: Let G be a split connected semisimple group over a field K. We give a conjectural formula for the motive of the stack of G-bundles over a curve C, in terms of special values of the motivic zeta function of C. The formula is true if C=P1 or G=SLn. If K=C, upon applying the PoincarŽ or Serre characteristic, the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If K=Fq, upon applying the counting measure, it reduces to the fact that the Tamagawa number of G over the function field of C is |¹1(G)|. This is joint work with Ajneet Dhillon.