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H 2 Algebraic cycles on varieties over finite fields

Auteurs : Pirutka, Alena (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... Chow groups cohomology groups computing cohomology cycle class maps Tate conjecture integral Tate conjecture topological obstructions algebraic obstructions unramified cohomology fibrations in quadrics cubic fourfolds Questions

Résumé : Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to understand. One then construct maps called "cycle class maps" from Chow groups to several cohomological theories.
In this talk, we focus on the case of a variety $X$ over a finite field. In this case, Tate conjecture claims the surjectivity of the cycle class map with rational coefficients; this conjecture is still widely open. In case of integral coefficients, we speak about the integral version of the conjecture and we know several counterexamples for the surjectivity. In this talk, we present a survey of some well-known results on this subject and discuss other properties of algebraic cycles which are either proved or expected to be true. We also discuss several involved methods.

Codes MSC :
14C15 - (Equivariant) Chow groups and rings; motives
14C25 - Algebraic cycles
14G15 - Finite ground fields
14H05 - Algebraic functions; function fields
14J70 - Algebraic hypersurfaces

 Informations sur la Vidéo Langue : Anglais Date de publication : 04/06/15 Date de captation : 18/05/15 Collection : Research talks ; Algebraic and Complex Geometry Format : QuickTime (.mov) Durée : 00:48:58 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2015-05-18_Pirutka.mp4 Informations sur la rencontre Nom de la rencontre : Arithmetics, geometry, cryptography and coding theory / Arithmétique, géométrie, cryptographie et théorie des codesOrganisateurs de la rencontre : Bassa, Alp ; Couvreur, Alain ; Kohel, DavidDates : 18/05/15 - 22/05/15 Année de la rencontre : 2015 URL Congrès : http://conferences.cirm-math.fr/1193.html Citation Data DOI : 10.24350/CIRM.V.18764303 Cite this video as: Pirutka, Alena (2015). Algebraic cycles on varieties over finite fields. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18764303 URI : http://dx.doi.org/10.24350/CIRM.V.18764303

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