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

Esnault, Hélène; Wittenberg, Olivier

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

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 $\ell$-adique entière pour de nombreuses surfaces de genre géométrique non nul, sur les corps locaux de caractéristique résiduelle $\ne \ell$. Le même énoncé vaut pour les surfaces $K3$ semi-stables définies sur $𝐂\left(\right(t\left)\right)$, 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 $\ell$-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic $\ne \ell$. The same statement holds for semistable $K3$ surfaces defined over $𝐂\left(\right(t\left)\right)$, but does not hold in general for surfaces over strictly local fields.

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

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