## Algebraic Geometry Seminar 2006

- Start Date: 04/24/2006
- End Date: 04/25/2006

Arvind Nair (Tata Institute for Fundamental Research, Mumbai, Maharashtra, India)

Elham Izadi (University of Georgia in Athens, Georgia, USA)

Kai Behrend (University of British Columbia)

University of British Columbia

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.