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.

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     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://archive.numdam.org/articles/10.24033/asens.2264/}
}

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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://archive.numdam.org/articles/10.24033/asens.2264/

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