On the cycle class map for zero-cycles over local fields

Esnault, Hélène; Wittenberg, Olivier

doi:10.24033/asens.2288

On the cycle class map for zero-cycles over local fields
[Sur l'application classe de cycle pour les zéro-cycles sur les corps locaux]

Esnault, Hélène ; Wittenberg, Olivier

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 483-520.

Résumé
Abstract

Nous étudions le groupe de Chow des 0-cycles des variétés projectives et lisses sur les corps locaux et strictement locaux. Nous prouvons en particulier l'injectivité de l'application classe de cycle vers la cohomologie $ℓ$ -adique entière pour de nombreuses surfaces de genre géométrique non nul, sur les corps locaux de caractéristique résiduelle $\neq ℓ$ . Le même énoncé vaut pour les surfaces $K 3$ semi-stables définies sur $𝐂 ((t))$ , mais ne vaut pas en général pour les surfaces sur les corps strictement locaux.

We study the Chow group of 0-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral $ℓ$ -adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic $\neq ℓ$ . The same statement holds for semistable $K 3$ surfaces defined over $𝐂 ((t))$ , but does not hold in general for surfaces over strictly local fields.

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2288

Classification : 14C15; 14G20, 14J28, 14D06
Keywords: Zero-cycles, local fields, K3 surfaces.
Mot clés : Zéro-cycles, corps locaux, surfaces K3.

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     author = {Esnault, H\'el\`ene and Wittenberg, Olivier},
     title = {On the cycle class map for zero-cycles over local fields},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {483--520},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {2},
     year = {2016},
     doi = {10.24033/asens.2288},
     mrnumber = {3481356},
     zbl = {1408.14015},
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     url = {https://www.numdam.org/articles/10.24033/asens.2288/}
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Esnault, Hélène; Wittenberg, Olivier. On the cycle class map for zero-cycles over local fields. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 2, pp. 483-520. doi : 10.24033/asens.2288. https://www.numdam.org/articles/10.24033/asens.2288/

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