Finiteness of K3 surfaces  and the Tate conjecture

Lieblich, Max; Maulik, Davesh; Snowden, Andrew

doi:10.24033/asens.2215

Finiteness of K3 surfaces and the Tate conjecture
[Finitude de surfaces K3 et la conjecture de Tate]

Lieblich, Max ; Maulik, Davesh ; Snowden, Andrew

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 285-308.

Résumé
Abstract

Étant donné un corps $k$ fini de caractéristique $p \geq 5$ , nous montrons que la conjecture de Tate pour les surfaces K3 sur $\bar{k}$ est vérifiée si et seulement s'il existe un nombre fini de surfaces K3 définies sur chaque extension finie de $k$ .

Given a finite field $k$ of characteristic $p \geq 5$ , we show that the Tate conjecture holds for K3 surfaces over $\bar{k}$ if and only if there are only finitely many K3 surfaces defined over each finite extension of $k$ .

Publié le : 2014-09-24

MR Zbl

DOI : 10.24033/asens.2215

Classification : 14G15, 14J28.
Keywords: Tate conjecture, twisted sheaves, K3 surfaces, Fourier-Mukai equivalence.
Mot clés : Conjecture de Tate, faisceaux tordus, surfaces K3, équivalence de Fourier-Mukai.

@article{ASENS_2014__47_2_285_0,
     author = {Lieblich, Max and Maulik, Davesh and Snowden, Andrew},
     title = {Finiteness of {K3} surfaces  and the {Tate} conjecture},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {285--308},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
     number = {2},
     year = {2014},
     doi = {10.24033/asens.2215},
     mrnumber = {3215924},
     zbl = {1329.14078},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2215/}
}

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Lieblich, Max; Maulik, Davesh; Snowden, Andrew. Finiteness of K3 surfaces  and the Tate conjecture. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 2, pp. 285-308. doi : 10.24033/asens.2215. https://www.numdam.org/articles/10.24033/asens.2215/

Bibliographie
Cité par

Artin, M. Supersingular K3 surfaces, Ann. Sci. École Norm. Sup., Volume 7 (1974), p. 543-567 (1975) (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Cassels, J. W. S., London Mathematical Society Monographs, 13, Academic Press Inc., 1978, 413 pages (ISBN: 0-12-163260-1) | MR | Zbl

Căldăraru, A. Nonfine moduli spaces of sheaves on K3 surfaces, Int. Math. Res. Not., Volume 2002 (2002), pp. 1027-1056 (ISSN: 1073-7928) | DOI | MR | Zbl

Deligne, P. La conjecture de Weil pour les surfaces K3, Invent. Math., Volume 15 (1972), pp. 206-226 (ISSN: 0020-9910) | DOI | MR | Zbl

Deligne, P., Algebraic surfaces (Orsay, 1976–78) (Lecture Notes in Math.), Volume 868, Springer, 1981, pp. 58-79 | MR | Zbl

Huybrechts, D.; Stellari, P. Equivalences of twisted K3 surfaces, Math. Ann., Volume 332 (2005), pp. 901-936 (ISSN: 0025-5831) | DOI | MR | Zbl

Lieblich, M. Moduli of twisted sheaves, Duke Math. J., Volume 138 (2007), pp. 23-118 (ISSN: 0012-7094) | DOI | MR | Zbl

Lieblich, M. Twisted sheaves and the period-index problem, Compos. Math., Volume 144 (2008), pp. 1-31 (ISSN: 0010-437X) | DOI | MR | Zbl

Lieblich, M. Moduli of twisted orbifold sheaves, Adv. Math., Volume 226 (2011), pp. 4145-4182 (ISSN: 0001-8708) | DOI | MR | Zbl

Liu, Q.; Lorenzini, D.; Raynaud, M. On the Brauer group of a surface, Invent. Math., Volume 159 (2005), pp. 673-676 (ISSN: 0020-9910) | DOI | MR | Zbl

Lieblich, M.; Maulik, D. A note on the cone conjecture for K3 surfaces in positive characteristic (preprint arXiv:1102.3377 ) | MR

Lieblich, M.; Olsson, M. Derived equivalence of K3 surfaces in positive characteristic (in preparation)

Lang, S.; Weil, A. Number of points of varieties in finite fields, Amer. J. Math., Volume 76 (1954), pp. 819-827 (ISSN: 0002-9327) | DOI | MR | Zbl

Milne, J. S. Abelian varieties (2008) (available at http://www.jmilne.org/math/CourseNotes/AV.pdf )

Mukai, S., Vector bundles on algebraic varieties (Bombay, 1984) (Tata Inst. Fund. Res. Stud. Math.), Volume 11, Tata Inst. Fund. Res., 1987, pp. 341-413 | MR | Zbl

Nygaard, N.; Ogus, A. Tate's conjecture for K3 surfaces of finite height, Ann. of Math., Volume 122 (1985), pp. 461-507 (ISSN: 0003-486X) | DOI | MR | Zbl

Ogus, A., Arithmetic and geometry, Vol. II (Progr. Math.), Volume 36, Birkhäuser, 1983, pp. 361-394 | DOI | MR | Zbl

Ogus, A. $F$ -isocrystals and de Rham cohomology. II. Convergent isocrystals, Duke Math. J., Volume 51 (1984), pp. 765-850 (ISSN: 0012-7094) | DOI | MR | Zbl

Saint-Donat, B. Projective models of K3 surfaces, Amer. J. Math., Volume 96 (1974), pp. 602-639 (ISSN: 0002-9327) | DOI | MR | Zbl

Serre, J.-P., Springer, 1973, 115 pages | MR | Zbl

Totaro, B., Current developments in algebraic geometry (Math. Sci. Res. Inst. Publ.), Volume 59, Cambridge Univ. Press, 2012, pp. 405-426 | MR | Zbl

Yoshioka, K., Moduli spaces and arithmetic geometry (Adv. Stud. Pure Math.), Volume 45, Math. Soc. Japan, 2006, pp. 1-30 | MR | Zbl

Zarhin, J. G. Endomorphisms of abelian varieties and points of finite order in characteristic $P$ , Mat. Zametki, Volume 21 (1977), pp. 737-744 (ISSN: 0025-567X) | MR | Zbl

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