H 2 Algebraic cycles on varieties over finite fields

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

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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 : Algebraic and Complex Geometry
    Sous collection : Research talks
    Domaine : Algebraic & Complex Geometry
    Format : QuickTime (.mov) Durée : 00:48:58
    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 codes
Organisateurs de la rencontre : Bassa, Alp ; Couvreur, Alain ; Kohel, David
Dates : 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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