## ECOVA - Etude COhomologique des Variétés Algébriques

Description du Projet: This project, at the crossroads of algebraic geometry and number theory, aims at investigating some aspects relating the arithmetico-geometric and cohomological properties of an algebraic variety. The notion of cohomology is somewhat the modern culmination of the old idea of linearization', which consists in associating to an object with a rich structure (an algebraic variety), families of simpler objects (linear i.e. vector spaces or modules with additional data - for instance a cohomology group) supposed to encode lots of the relevant information about the original object but easier to handle and classify. Abelian cohomology has progressively emerged in topology and complex geometry during the first half of the twentieth century. During this period, first, and second cohomology pointed sets also appear with the study of twists in Galois theory. In the fifties and early sixties, the development of category theory and homological algebra provided the conceptual frame to elaborate systematically a cohomological formalism in abelian categories, paving the way to its diffusion in many areas of mathematics. In particular, it is intrinsically connected to the breathtaking development of modern algebraic geometry, yielding to the introduction of various algebraic Weil cohomologies related by comparison isomorphisms. Around the early seventies, these several cohomological avatars were unified in Grothendieck's fascinating conjectural theory of pure motives, which was later included in the wider theory of mixed motives. These theories - partly conjectural - naturally give rise to conjectures horizon', such as the Hodge and the Tate conjectures, various conjectures on algebraic cycles etc. The purpose of ECOVA is to study a series of problems arising as consequences or particular cases of these conjectures horizon', which can be roughly classified according to the three following main directions:
• Complex and l-adic coefficients: absolute setting (special cases of the Hodge and the Tate conjectures, Mumford-Tate conjecture), variational setting (geometric study of the exceptional loci, representations of the fundamental group, l-independence, Shimura varieties);
• Integral, finite, adelic coefficients: absolute setting (obstructions to the integral variant of the Hodge and the Tate conjectures), variational setting (representations of the fundamental group, semi simplicity modulo l, adelic openness and l-independence and Shimura varieties);
• Finer questions on algebraic cycles (Higher Abel-Jacobi maps, integral and birational aspects, study of zero-cycles and rational points, mixed motives).

Référence du projet: ANR-15-CE40-0002-01
Document scientifique

Membres

Olivier Benoist
François Charles
Cyril Demarche
Lie Fu
Bruno Klingler
Baptiste Morin
Fabrice Orgogozo
Alena Pirutka
Joël Riou
Olivier Wittenberg

Workshops et conférences

Rencontre d'ouverture E.N.S. Paris (Salle W) - 10 février 2015 (Informations).

Conférence Cohomology of Algebraic varieties', C.I.R.M., 26-30 mars 2018.

Groupes de travail, cours etc.

• Cours de Matthew Morrow: Integral p-adic Hodge theory

Jussieu, salle 1516-413: lundi 18, 25 janvier et 1er février 2015 de 10h30 à 12h30
Jussieu, salle 1516-101: vendredi 25, 22 et 29 janvier 2015 de 10h30 à 12h30

• Groupe de travail: Faisceaux lisses et isocristaux: finitude et compagnons

Rencontre 1: 20 - 21 avril 2017, IHP

20 avril 2017, IHP - Salle 201

14:00-16:00: 2.(3) Premières propriétés de $\ell$-indépendance dans un système compatible (E. Ambrosi)
16:30-18:00: 3.(1) Conjectures de Weil - première partie (O. Benoist)

21 avril, IHP - Salle 314

9:30-10:30: 3.(1) Conjectures de Weil - deuxième partie (O. Benoist) Notes
11:00-13:00: 3.(2) Semisimplicité et $\ell$-indépendance dans un système compatible pur rationnel en caractéristique positive (M. D'Addezio)

Rencontre 2: 30 - 31 mai 2017, IHP

30 mai 2017, IHP - Salle 201

9:30-10:30: 4.(1) Théorèmes de Bertini/irréductibilité de Hilbert et pureté (A. Cadoret)
11:00-12:30: 4.(2) Algebraicité I (F. Orgogozo)
13:30-15:00: 4.(2) Algebraicité I (F. Orgogozo)
15:30-17:00: 4.(3) Théorème de reconstruction de Drinfeld et existence de compagnons I (J. Riou)

31 mai 2017, IHP - Salle 201

9:30-11:00: 4.(3) Théorème de reconstruction de Drinfeld et existence de compagnons II (J. Riou)
11:30-13:00: 5. Modules et finitude I (J-B Teyssier)
14:00-15:30: 5. Modules et finitude II (J-B Teyssier)

Autres Événements scientifiques coorganisés par les membres du projet

Séminaires

Autour des Cycles algébriques
Variétés rationnelles

Workshops et conférences

Positivity in arithmetic and geometry, Orsay, 29/05-02/06/2017
Arithmetic algebraic geometry, N.Y.U., 29/08-02/09/2016
Etale fundamental groups in arithmetic geometry, I.H.P. et E.N.S., 27/05-03/06/2016
Sino-French workshop in arithmetic and algebraic geometry, Bordeaux, 24-28/05/2016