Fourier-Mukai partners of K3 surfaces  in positive characteristic

Lieblich, Max; Olsson, Martin

doi:10.24033/asens.2264

Fourier-Mukai partners of K3 surfaces in positive characteristic
[Compagnons de Fourier-Mukai des surfaces K3 en caractéristique positive]

Lieblich, Max ; Olsson, Martin

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1001-1033.

Résumé
Abstract

Nous étudions les équivalences de Fourier-Mukai entre surfaces de type K3 en caractéristique positive et démontrons que les résultats classiques sur les complexes se généralisent sans modifications. Le résultat clef est un “théorème de Torelli” pour les catégories dérivées. Comme conséquence, toute K3 surface supersingulière est determinée uniquement à isomorphisme près par sa catégorie dérivée. Nous étudions de plus quelques réalisations algébriques de structure de Mukai-Hodge et les utilisons pour prouver que : 1) la fonction zêta d'une surface de type K3 est une invariante dérivée (découverte indépendamment par Huybrechts) ; 2) la conjecture variationnelle cristalline de Hodge est vérifiée pour les correspondances entre produits de surfaces de type K3 résultant de transformés de Fourier-Mukai.

We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the classical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hodge structure in algebraic cohomology theories (étale, crystalline, de Rham) and use these to prove: 1) the zeta function of a K3 surface is a derived invariant (discovered independently by Huybrechts); 2) the variational crystalline Hodge conjecture holds for correspondences arising from Fourier-Mukai kernels on products of two K3 surfaces.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2264

Classification : 14J28; 14G17.
Keywords: Fourier-Mukai equivalence, K3 surfaces, zeta function, motives.
Mot clés : Équivalence de Fourier-Mukai, surfaces de type K3, fonctions zêta, motifs.

@article{ASENS_2015__48_5_1001_0,
     author = {Lieblich, Max and Olsson, Martin},
     title = {Fourier-Mukai partners of {K3} surfaces  in positive characteristic},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1001--1033},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {5},
     year = {2015},
     doi = {10.24033/asens.2264},
     mrnumber = {3429474},
     zbl = {1342.14038},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2264/}
}

TY  - JOUR
AU  - Lieblich, Max
AU  - Olsson, Martin
TI  - Fourier-Mukai partners of K3 surfaces  in positive characteristic
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2015
SP  - 1001
EP  - 1033
VL  - 48
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2264/
DO  - 10.24033/asens.2264
LA  - en
ID  - ASENS_2015__48_5_1001_0
ER  -

%0 Journal Article
%A Lieblich, Max
%A Olsson, Martin
%T Fourier-Mukai partners of K3 surfaces  in positive characteristic
%J Annales scientifiques de l'École Normale Supérieure
%D 2015
%P 1001-1033
%V 48
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2264/
%R 10.24033/asens.2264
%G en
%F ASENS_2015__48_5_1001_0

Lieblich, Max; Olsson, Martin. Fourier-Mukai partners of K3 surfaces  in positive characteristic. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1001-1033. doi : 10.24033/asens.2264. http://www.numdam.org/articles/10.24033/asens.2264/

Bibliographie
Cité par

Bridgeland, T.; Maciocia, A. Complex surfaces with equivalent derived categories, Math. Z., Volume 236 (2001), pp. 677-697 (ISSN: 0025-5874) | DOI | MR | Zbl

Berthelot, P.; Ogus, A. $F$ -isocrystals and de Rham cohomology. I, Invent. Math., Volume 72 (1983), pp. 159-199 (ISSN: 0020-9910) | DOI | MR | Zbl

Bridgeland, T. Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc., Volume 31 (1999), pp. 25-34 (ISSN: 0024-6093) | DOI | MR | Zbl

Charles, F. The Tate conjecture for $K 3$ surfaces over finite fields, Invent. Math., Volume 194 (2013), pp. 119-145 (ISSN: 0020-9910) | DOI | MR | Zbl

Canonaco, A.; Stellari, P. Twisted Fourier-Mukai functors, Adv. Math., Volume 212 (2007), pp. 484-503 (ISSN: 0001-8708) | DOI | MR | Zbl

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

Dieudonné, J.; Grothendieck, A., Publ. Math. IHÉS, 4, 8, 11, 17, 20, 24, 28, 32, 1961–1967 | Numdam | MR | Zbl

Gillet, H.; Messing, W. Cycle classes and Riemann-Roch for crystalline cohomology, Duke Math. J., Volume 55 (1987), pp. 501-538 (ISSN: 0012-7094) | DOI | MR | Zbl

Gros, M. Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique, Mém. Soc. Math. France, Volume 21 (1985) (ISSN: 0037-9484) | Numdam | MR | Zbl

Huybrechts, D.; Lehn, M., Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 2010, 325 pages (ISBN: 978-0-521-13420-0) | DOI | MR | Zbl

Hosono, S.; Lian, B. H.; Oguiso, K.; Yau, S.-T., Algebraic structures and moduli spaces (CRM Proc. Lecture Notes), Volume 38, Amer. Math. Soc., Providence, RI, 2004, pp. 177-192 | DOI | MR | Zbl

Huybrechts, D., Oxford Mathematical Monographs, The Clarendon Press, Oxford Univ. Press, Oxford, 2006, 307 pages (ISBN: 978-0-19-929686-6; 0-19-929686-3) | DOI | MR | Zbl

Huybrechts, D. Derived and abelian equivalence of $K 3$ surfaces, J. Algebraic Geom., Volume 17 (2008), pp. 375-400 (ISSN: 1056-3911) | DOI | MR | Zbl

Illusie, L., Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43-70 | DOI | MR | Zbl

Kleiman, S. L., Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3-20 | DOI | MR | Zbl

Langer, A. Semistable sheaves in positive characteristic, Ann. of Math., Volume 159 (2004), pp. 251-276 (ISSN: 0003-486X) | DOI | MR | Zbl

Langton, S. G. Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math., Volume 101 (1975), pp. 88-110 (ISSN: 0003-486X) | DOI | MR | Zbl

Laumon, G., Séminaire de géométrie analytique (École Norm. Sup., Paris, 1974-75) (Astérisque), Volume 36–37, Soc. Math. France, Paris, 1976, pp. 163-188 | Numdam | MR | Zbl

Lieblich, M. Moduli of complexes on a proper morphism, J. Algebraic Geom., Volume 15 (2006), pp. 175-206 (ISSN: 1056-3911) | 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, Compositio Math., Volume 144 (2008), pp. 1-31 (ISSN: 0010-437X) | DOI | MR | Zbl

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

Matsusaka, T.; Mumford, D. Two fundamental theorems on deformations of polarized varieties, Amer. J. Math., Volume 86 (1964), pp. 668-684 (ISSN: 0002-9327) | DOI | MR | Zbl

Maulik, D.; Poonen, B. Néron-Severi groups under specialization, Duke Math. J., Volume 161 (2012), pp. 2167-2206 (ISSN: 0012-7094) | DOI | MR | Zbl

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

Ogus, A., Moduli of abelian varieties (Texel Island, 1999) (Progr. Math.), Volume 195, Birkhäuser, 2001, pp. 325-343 | DOI | MR | Zbl

Ogus, A., Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II (Astérisque), Volume 64, Soc. Math. France, Paris, 1979, pp. 3-86 | Numdam | MR | Zbl

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

Orlov, D. O. Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, Volume 58 (2003), pp. 89-172 ; translation: Russian Math. Surveys 58 (2003), 511–591 (ISSN: 0042-1316) | DOI | MR | Zbl

Orlov, D. O. Equivalences of derived categories and $K 3$ surfaces, J. Math. Sci., Volume 84 (1997), pp. 1361-1381 (ISSN: 1072-3374) | DOI | MR | Zbl

Pera, K. M. The Tate conjecture for K3 surfaces in odd characteristic (preprint arXiv:1301.6326 ) | MR

Petrequin, D. Classes de Chern et classes de cycles en cohomologie rigide, Bull. Soc. Math. France, Volume 131 (2003), pp. 59-121 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Weibel, C. A., Cambridge Studies in Advanced Math., 38, Cambridge Univ. Press, Cambridge, 1994, 450 pages (ISBN: 0-521-43500-5; 0-521-55987-1) | DOI | MR | Zbl

Cité par Sources :