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|Orateur(s)||Thomas Krämer, Morten Lüders et Timo Richarz - ,|
|Titre||Séminaire Autour des cycles algébriques|
|Horaire||14:00 à 16:00|
|Résume||14h00--15h00 : Morten Lüders (IMJ-PRG) Local to global principles for higher zero-cycles|
We explain the relationship between the Tate-Poitou exact sequence, the Kato conjectures and local to global principles for higher Chow groups for smooth projective schemes over global fields. This is joint work with Johann Haas.
15h30--16h30 : Timo Richarz (Technische Universität Darmstadt) The intersection motive of the moduli stack of shtukas
Moduli stacks of shtukas are regarded as function field analogues of Shimura varieties, and their étale cohomology is known to realize the Langlands correspondence for these fields. For the general linear group such a correspondence was established by L. Lafforgue in the 90’s building upon earlier work of Drinfeld. In a recent breakthrough V. Lafforgue constructs the Automorphic to Galois direction of the correspondence for general reductive groups G over function fields. His completely new method makes it possible to systematically analyze the requirements of the cohomology theory needed in order to establish such a correspondence.
In the talk I report on joint work with J. Scholbach which aims at applying the theory of motives as developed by Voevodsky, Levine, Hanamura, Ayoub, Cisinski-Déglise and many others to the constructions in the work of V. Lafforgue. As a first step we show that the intersection (cohomology) motive of the moduli stack of G-shtukas is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. Also we establish the analogue of the Geometric Satake Isomorphism of Lusztig, Ginzburg and Mirkovic-Vilonen in this set-up. This is in accordance with general expectations on the independence of $\ell$ in the Langlands correspondence for function fields.
17h00--18h00 : Thomas Krämer (Humboldt-Universität zu Berlin) A converse to Riemann's theorem on Jacobian varieties
Jacobians of curves have been studied a lot since Riemann's theorem, which says that their theta divisor is a sum of copies of the curve. Similarly, for intermediate Jacobians of smooth cubic threefolds Clemens and Griffiths showed that the theta divisor is a sum of two copies of the Fano surface of lines on the threefold. We prove that in both cases these are the only decompositions of the theta divisor, extending previous results of Casalaina-Martin, Popa and Schreieder. Our ideas apply to a much wider context and only rely on the decomposition theorem for perverse sheaves and some representation theory.
(À noter : aucun séminaire en avril.)