Paris Diderot Sorbonne Université CNRS

Autour des cycles algébriques

2017 2018 2019

Année 2018- 2019

Organisateurs : A. Cadoret - F. Charles - J. Fresán - M. Morrow

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Thomas Krämer, Morten Lüders et Timo Richarz

Séminaire Autour des cycles algébriques

mercredi 15 mai 2019 à 14:00 : Jussieu, couloir 15-16, salle 413

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.)

Martin Gallauer, Quentin Guignard et Lucia Mocz

Séminaire Autour des cycles algébriques

mercredi 20 mars 2019 à 14:00 : Jussieu, couloir 15-16, salle 413

14h00—15h00 : Martin Gallauer (University of Oxford) How many real Artin-Tate motives are there ?
The goals of my talk are 1) to place this question within the framework of tensor-triangular geometry, and 2) to report on joint work with Paul Balmer (UCLA) which provides an answer in this framework.

15h30—16h30 : Quentin Guignard (ENS) Facteurs locaux géométriques
On commencera par rappeler la théorie (due à Deligne-Langlands) des facteurs epsilon de représentations ℓ-adiques sur des corps locaux à corps résiduel fini. On expliquera ensuite comment définir des facteurs locaux pour des représentations ℓ-adiques sur des corps de séries de Laurent k((t)), avec k un corps parfait de caractéristique positive p différente de ℓ. On dispose alors d’une décomposition du déterminant de la cohomologie d’un faisceau ℓ-adique sur une courbe lisse sur un tel corps k, en un produit de contributions locales. Lorsque k est fini, on retrouve ainsi la théorie classique de Dwork, Deligne, Langlands et Laumon.

17h00—18h00 : Lucia Mocz (Universität Bonn) Harder-Narasimhan Theory and Faltings Height
We expand on the relation between the Faltings height and Harder-Narasimhan theory for finite flat group schemes which was originally observed by Fargues and used in the proof of the CM Northcott Property for the Faltings height. We will also demonstrate in the developments of this theory a particular pathological behavior of the Faltings height, and draw connections to other conjectures.

Marco d'Addezio et Dimitri Wyss

Séminaire Autour des cycles algébriques

mercredi 20 février 2019 à 14:30 : Jussieu, couloir 15-16, salle 413

14h30—15h30 : Dimitri Wyss (IMJ-PRG) Volume non-archimédien de la fibration de Hitchin
Hausel et Thaddeus on conjecturé une égalité entre les nombres de Hodge ’stringy’ des espaces de Hitchin pour les groupes SL_n et PGL_n. Motivé par cette conjecture on étudie l’intégration non-archimedienne sur la fibration de Hitchin dans le sense de Denef-Loeser et Batyrev. En utilisant la dualité des fibres de Hitchin générique on arrive ainsi à démontrer la conjecture de Hausel-Thaddeus. Dans un contexte plus arithmétique les mêmes idées donnent une nouvelle preuve de la stabilisation géométrique pour les fibres de Hitchin anisotropes, un énoncé clé dans la preuve du lemme fondamental par Ngô. C’est un travail en commun avec Michael Groechenig et Paul Ziegler.

16h00—17h00 : Marco d’Addezio (Freie Universität, Berlin) Finiteness of perfect torsion points of an abelian variety and F-isocrystals
I will report on a joint work with Emiliano Ambrosi. Let k be a field which is finitely generated over the algebraic closure of a finite field. As a consequence of the theorem of Lang-Néron, for every abelian variety over k which does not admit any isotrivial abelian subvariety, the group of k-rational torsion points is finite. We show that the same is true for the group of torsion points defined on a perfect closure of k. This gives a positive answer to a question posed by Hélène Esnault in 2011. To prove the theorem we translate the problem into a certain question on morphisms of F-isocrystals. Then we handle it by studying the monodromy groups of the F-isocrystals involved.

Dustin Clausen, Frédéric Déglise et Thomas Nikolaus

Séminaire Autour des cycles algébriques

mercredi 21 novembre 2018 à 14:00 : (ATTENTION : SALLE INHABITUELLE) Jussieu, couloir 15-25, salle 502

14h00—15h00 : Frédéric Déglise (Université de Bourgogne et CNRS) t-structures on motives and vanishing
The motivic t-structure on Voevodsky mixed motives is the principal remaining problem of the theory envisioned by Beilinson. A by-product of the theory defined by Voevodsky over a perfect field is the existence of a natural t-structure, the homotopy t-structure. Surprisingly the difference between the homotopy t-structure and the hoped-for motivic t-structure is a priori very small ; just one misplaced shift. But the heart of the homotope t-structure is already far from being the category of abelian mixed motives.
One the other hand, the homotopy t-structure has good properties and is amenable to computation. In particular, it can be glued other arbitrary bases, much like the perverse homotopy t-structure. The resulting t-structure was first studied by Ayoub. As we have shown with Bondarko, a major tool in this regard is the use of dimension functions, in close analogy to Gabber’s treatment of perverse t-structures. This allows us to improve the results of Ayoub, and give new computations, sometime close to what exists for perverse sheaves.
This leads us to think that the study of the homotopy t-structure might contain the key to establish the existence of the motivic t-structure. Looking back to an old tentative definition of Voevodsky, I will illustrate this feeling by renewing two conjectures of the field, showing that they act as a bridge between the homotopy and motivic t-structures.

15h30—16h30 : Dustin Clausen (Universität Bonn) On Etale K-theory
In general, the etale sheafification operation on presheaves of spectra is completely inexplicit, and etale sheaves of spectra do not have the same good formal properties as etale cohomology does. Nonetheless we show that these troubles don’t arise for the etale sheafification of algebraic K-theory, which has surprisingly good properties, including : 1) continuity ; 2) hidden functoriality ; 3) a strong connection with Weil-style cohomology theories via a "motivic filtration". Furthermore, all of this is valid over a completely general commutative ring. This is joint work with Akhil Mathew, in parts also with Bharghav Bhatt.

17h00—18h00 : Thomas Nikolaus (Universität Münster) Cyclotomic spectra and Cartier modules
We review the notion of cyclotomic spectra and basic examples such as THH. Then we explain how this is related by a t-structure to (integral, p-typical) Cartier modules. This explains nicely the appearance of de Rham Witt complexes in the theory. The comparison relies on the notion of a topological Cartier module. We present examples such as Witt vectors and K-theory of endomorphisms. This is joint work with Ben Antieau.