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.

@article{ASENS_2016__49_2_483_0,
     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},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2288/}
}

TY  - JOUR
AU  - Esnault, Hélène
AU  - Wittenberg, Olivier
TI  - On the cycle class map for zero-cycles over local fields
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 483
EP  - 520
VL  - 49
IS  - 2
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://archive.numdam.org/articles/10.24033/asens.2288/
DO  - 10.24033/asens.2288
LA  - en
ID  - ASENS_2016__49_2_483_0
ER  -

%0 Journal Article
%A Esnault, Hélène
%A Wittenberg, Olivier
%T On the cycle class map for zero-cycles over local fields
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 483-520
%V 49
%N 2
%I Société Mathématique de France. Tous droits réservés
%U http://archive.numdam.org/articles/10.24033/asens.2288/
%R 10.24033/asens.2288
%G en
%F ASENS_2016__49_2_483_0

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. http://archive.numdam.org/articles/10.24033/asens.2288/

Bibliographie
Cité par

Abramovich, D.; Karu, K.; Matsuki, K.; Włodarczyk, J. Torification and factorization of birational maps, J. Amer. Math. Soc., Volume 15 (2002), pp. 531-572 | DOI | MR | Zbl

Anantharaman, S. Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1, Mém. Soc. Math. France, Volume 33 (1973), pp. 5-79 | DOI | Numdam | Zbl

Artin, M.; Swinnerton-Dyer, H. P. F. The Shafarevich-Tate conjecture for pencils of elliptic curves on $K 3$ surfaces, Invent. math., Volume 20 (1973), pp. 249-266 | DOI | MR | Zbl

Bloch, S.; Kas, A.; Lieberman, D. Zero cycles on surfaces with $p_{g} = 0$ , Compositio Math., Volume 33 (1976), pp. 135-145 | Numdam | MR | Zbl

Bloch, S. Torsion algebraic cycles, $K_{2}$ , and Brauer groups of function fields, Bull. Amer. Math. Soc., Volume 80 (1974), pp. 941-945 | DOI | MR | Zbl

Bloch, S., Duke University Mathematics Series, IV, Duke University Mathematics Department, 1980 | MR

Bloch, S. On the Chow groups of certain rational surfaces, Ann. Sci. École Norm. Sup., Volume 14 (1981), pp. 41-59 | DOI | Numdam | MR | Zbl

Bogomolov, F. A.; Zarhin, Y. G. Ordinary reduction of $K 3$ surfaces, Cent. Eur. J. Math., Volume 7 (2009), pp. 206-213 | MR | Zbl

Cossec, F. R.; Dolgachev, I. V., Progress in Math., 76, Birkhäuser, 1989, 397 pages | MR | Zbl

Colliot-Thélène, J.-L. Groupe de Chow des zéro-cycles sur les variétés $p$ -adiques (d'après S. Saito, K. Sato et al.), Séminaire Bourbaki, vol. 2009/2010, exposé n^o 1012, Astérisque, Volume 339 (2011), pp. 1-30 | Numdam | MR | Zbl

Colliot-Thélène, J.-L., Arithmetic geometry (Lecture Notes in Math.), Volume 2009, Springer, 2011, pp. 1-44 | DOI | MR | Zbl

Colliot-Thélène, J.-L. Hilbert's Theorem 90 for $K_{2}$ , with application to the Chow groups of rational surfaces, Invent. math., Volume 71 (1983), pp. 1-20 | DOI | MR | Zbl

Colliot-Thélène, J.-L., Arithmetic algebraic geometry (Trento, 1991) (Lecture Notes in Math.), Volume 1553, Springer, 1993, pp. 1-49 | MR | Zbl

Colliot-Thélène, J.-L.; Raskind, W. $𝒦_{2}$ -cohomology and the second Chow group, Math. Ann., Volume 270 (1985), pp. 165-199 | DOI | MR | Zbl

Colliot-Thélène, J.-L.; Raskind, W. Groupe de Chow de codimension deux des variétés définies sur un corps de nombres: un théorème de finitude pour la torsion, Invent. math., Volume 105 (1991), pp. 221-245 | DOI | MR | Zbl

Colliot-Thélène, J.-L.; Sansuc, J.-J. On the Chow groups of certain rational surfaces: a sequel to a paper of S. Bloch, Duke Math. J., Volume 48 (1981), pp. 421-447 | MR | Zbl

Colliot-Thélène, J.-L.; Saito, S. Zéro-cycles sur les variétés $p$ -adiques et groupe de Brauer, Int. Math. Res. Not., Volume 1996 (1996), pp. 151-160 | DOI | MR | Zbl

Colliot-Thélène, J.-L.; Voisin, C. Cohomologie non ramifiée et conjecture de Hodge entière, Duke Math. J., Volume 161 (2012), pp. 735-801 | MR | Zbl

de Jong, A. J. Smoothness, semi-stability and alterations, Publ. Math. de l'IHÉS, Volume 83 (1996), pp. 51-93 | DOI | Numdam | MR | Zbl

Deligne, P., Lecture Notes in Math., 569, Springer, 1977 (Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 $\frac{1}{2}$ ) | MR | Zbl

Deligne, P. La conjecture de Weil : II, Publ. Math. de l'IHÉS, Volume 52 (1980), pp. 137-252 | DOI | Numdam | MR | Zbl

Dubois, P.; Jarraud, P. Une propriété de commutation au changement de base des images directes supérieures du faisceau structural, C. R. Acad. Sci. Paris Sér. A, Volume 279 (1974), pp. 745-747 | MR | Zbl

Esnault, H.; Levine, M.; Wittenberg, O. Index of varieties over Henselian fields and Euler characteristic of coherent sheaves, J. Alg. Geom., Volume 24 (2015), pp. 693-718 | DOI | MR | Zbl

Freitag, E.; Kiehl, R., Ergebn. Math. Grenzg., 3. Folge, 13, Springer, 1988, 317 pages | MR | Zbl

Friedman, R.; Morrison, D. R., The birational geometry of degenerations (Cambridge, Mass., 1981) (Progr. Math.), Volume 29, Birkhäuser, 1983, pp. 1-32 | MR | Zbl

Friedman, R. Global smoothings of varieties with normal crossings, Ann. of Math., Volume 118 (1983), pp. 75-114 | DOI | MR | Zbl

Fujiwara, K., Algebraic geometry 2000, Azumino (Hotaka) (Adv. Stud. Pure Math.), Volume 36, Math. Soc. Japan, 2002, pp. 153-183 | DOI | MR | Zbl

Fulton, W., Ergebn. Math. Grenzg., 3. Folge, 2, Springer, 1998, 470 pages | MR | Zbl

Gabber, O.; Liu, Q.; Lorenzini, D. The index of an algebraic variety, Invent. math., Volume 192 (2013), pp. 567-626 | DOI | MR | Zbl

Guillén, F.; Navarro Aznar, V. Sur le théorème local des cycles invariants, Duke Math. J., Volume 61 (1990), pp. 133-155 | DOI | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie, Publ. Math. de l'IHÉS, Volume 24 (1965) | Numdam | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie, Publ. Math. de l'I.H.É.S., Volume 28 (1966) | Numdam | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, Publ. Math. de l'IHÉS, Volume 32 (1967) | Numdam | MR | Zbl

Grothendieck, A., Dix Exposés sur la Cohomologie des Schémas, North-Holland, 1968, pp. 88-188 | MR | Zbl

Grothendieck, A., Lecture Notes in Math., 288, Springer, 1972, 523 pages Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I) | Zbl

Grothendieck, A., Lecture Notes in Math., 305, Springer, 1973, 640 pages Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4) | Zbl

Gurjar, R. V.; Shastri, A. R. Covering spaces of an elliptic surface, Compositio Math., Volume 54 (1985), pp. 95-104 | Numdam | MR | Zbl

Hartshorne, R., 52, Springer, 1977, 496 pages (Graduate Texts in Mathematics) | MR | Zbl

Illusie, L. On Gabber's refined uniformization (talks at the Univ. Tokyo, Jan. 17, 22, 31, Feb. 7, 2008, available at http://www.math.u-psud.fr/~illusie/refined_uniformization3.pdf )

Illusie, L. Autour du théorème de monodromie locale, Astérisque, Volume 223 (1994), pp. 9-57 Périodes $p$ -adiques (Bures-sur-Yvette, 1988) | Numdam | MR | Zbl

Ito, T. Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math., Volume 127 (2005), pp. 647-658 | DOI | MR | Zbl

Jouanolou, J.-P., Progress in Math., 42, Birkhäuser, 1983, 127 pages | MR | Zbl

Kleiman, S.; Altman, A. Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra, Volume 7 (1979), pp. 775-790 | DOI | MR | Zbl

Kleiman, S., Exp. XIII, Théorie des intersections et théorème de Riemann–Roch (Lecture Notes in Math.), Volume 225, Springer, 1971 Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6)

Kodaira, K. On compact analytic surfaces: II, Ann. of Math., Volume 77 (1963), pp. 563-626 | DOI | MR | Zbl

Kollár, J., Higher-dimensional geometry over finite fields (NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur.), Volume 16, IOS, 2008, pp. 92-122 (Notes by Ulrich Derenthal) | MR | Zbl

Kollár, J. Higher direct images of dualizing sheaves I, Ann. of Math., Volume 123 (1986), pp. 11-42 | DOI | MR | Zbl

Kollár, J. Rationally connected varieties over local fields, Ann. of Math., Volume 150 (1999), pp. 357-367 | DOI | MR | Zbl

Kollár, J.; Szabó, E. Rationally connected varieties over finite fields, Duke Math. J., Volume 120 (2003), pp. 251-267 | DOI | MR | Zbl

Kato, K.; Saito, S. Unramified class field theory of arithmetical surfaces, Ann. of Math., Volume 118 (1983), pp. 241-275 | DOI | MR | Zbl

Kulikov, V. S. Degenerations of $K 3$ surfaces and Enriques surfaces, Izv. Akad. Nauk SSSR Ser. Mat., Volume 41 (1977), pp. 1008-1042 English translation: Math. USSR-Izv. 11 (1977), no. 5, 957–989 | MR | Zbl

Laza, R. Triangulations of the sphere and degenerations of $K 3$ surfaces (2008) (preprint)

Lichtenbaum, S. Duality theorems for curves over $p$ -adic fields, Invent. math., Volume 7 (1969), pp. 120-136 | DOI | MR | Zbl

Looijenga, E. Rational surfaces with an anticanonical cycle, Ann. of Math., Volume 114 (1981), pp. 267-322 | DOI | MR | Zbl

Lang, S.; Tate, J. Principal homogeneous spaces over abelian varieties, Amer. J. Math., Volume 80 (1958), pp. 659-684 | DOI | MR | Zbl

Manin, Y. I., Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, 1971, pp. 401-411 | MR | Zbl

Milne, J. S., BookSurge, LLC, Charleston, SC, 2006, 339 pages | MR | Zbl

Milne, J. S., Princeton Mathematical Series, 33, Princeton Univ. Press, 1980, 323 pages | MR | Zbl

Miranda, R.; Morrison, D. R., The birational geometry of degenerations (Cambridge, Mass., 1981) (Progr. Math.), Volume 29, Birkhäuser, 1983, pp. 173-259 | MR | Zbl

Mustaţă, M.; Nicaise, J. Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton, Algebr. Geom., Volume 2 (2015), pp. 365-404 (ISSN: 2214-2584) | DOI | MR | Zbl

Ogg, A. P. Cohomology of abelian varieties over function fields, Ann. of Math., Volume 76 (1962), pp. 185-212 | DOI | MR | Zbl

Persson, U. On degenerations of algebraic surfaces, Mem. Amer. Math. Soc., Volume 11 (1977) | MR | Zbl

Persson, U.; Pinkham, H. Degeneration of surfaces with trivial canonical bundle, Ann. of Math., Volume 113 (1981), pp. 45-66 | DOI | MR | Zbl

Parimala, R.; Suresh, V. Zero-cycles on quadric fibrations: finiteness theorems and the cycle map, Invent. math., Volume 122 (1995), pp. 83-117 | DOI | MR | Zbl

Poonen, B.; Voloch, J. F., Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002) (Progr. Math.), Volume 226, Birkhäuser, 2004, pp. 175-184 (with appendices by Jean-Louis Colliot-Thélène and Nicholas M. Katz) | DOI | MR | Zbl

Rapoport, M.; Zink, T. Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. math., Volume 68 (1982), pp. 21-101 | DOI | MR | Zbl

Saito, T. Weight spectral sequences and independence of $ℓ$ , J. Inst. Math. Jussieu, Volume 2 (2003), pp. 583-634 | DOI | MR | Zbl

Saito, S., Algebraic $K$ -theory: connections with geometry and topology (Lake Louise, AB, 1987) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.), Volume 279, Kluwer Acad. Publ., 1989, pp. 425-444 | MR | Zbl

Saito, S. On the cycle map for torsion algebraic cycles of codimension two, Invent. math., Volume 106 (1991), pp. 443-460 | DOI | MR | Zbl

Sato, K. Non-divisible cycles on surfaces over local fields, J. Number Theory, Volume 114 (2005), pp. 272-297 | DOI | MR | Zbl

Sato, K. Injectivity of the torsion cycle map of codimension two of varieties over $p$ -adic fields with semi-stable reduction, J. reine angew. Math., Volume 501 (1998), pp. 221-235 | DOI | MR | Zbl

Silverman, J. H., Graduate Texts in Math., 151, Springer, 1994, 525 pages | MR | Zbl

Saito, S.; Sato, K. A finiteness theorem for zero-cycles over $p$ -adic fields, Ann. of Math., Volume 172 (2010), pp. 1593-1639 | DOI | MR | Zbl

Saito, S.; Sato, K. Zero-cycles on varieties over $p$ -adic fields and Brauer groups, Ann. Sci. Éc. Norm. Sup., Volume 47 (2014), pp. 505-537 | DOI | Numdam | MR | Zbl

Steenbrink, J. Limits of Hodge structures, Invent. math., Volume 31 (1975/76), pp. 229-257 | DOI | MR | Zbl

Steenbrink, J. H. M., Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 525-563 | MR | Zbl

Tate, J., Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., 1994, pp. 71-83 | MR | Zbl

Yamazaki, T. On Chow and Brauer groups of a product of Mumford curves, Math. Ann., Volume 333 (2005), pp. 549-567 | DOI | MR | Zbl

Cité par Sources :