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Annales scientifiques de l'ENS - Parutions - série 4, 47 (2014)

Parutions < série 4, 47

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 47, fascicule 5 (2014)

Zsolt Patakfalvi
Semi-positivity in positive characteristics
Annales scientifiques de l'ENS 47, fascicule 5 (2014), 991-1025

Télécharger cet article : Fichier PDF

Résumé :
Semi-positivité en caractéristiques positives
Soit f: (X, ) Y une famille projective plate de paires nettement F-pures et log-canoniquement polarisées sur un corps algébriquement clos de caractéristique p> 0 tel que p ind (K_ X / Y + ). Nous montrons que K_X / Y + est nef et que f_* (O_X (m (K_X / Y + ))) est un fibré vectoriel nef pour m 0 et qu'il est assez divisible. Certains des résultats s'étendent également aux couples non log-canoniquement polarisés. La principale motivation de ces résultats est la projectivité de sous-espaces propres de l'espace des modules des paires stables en caractéristiques positives. D'autres applications incluent des nouvelles preuves algébriques des résultats de positivité en caractéristique nulle, et un cas particulier de sous-additivité de la dimension de Kodaira de caractéristique positive.

Mots-clefs : Semi-positif, nef, faisceau canonique relatif, nettement F-pur, fortement F-régulier, variétés stables, sous-additivité de la dimension de Kodaira.

Abstract:
Let f : (X, ) Y be a flat, projective family of sharply F-pure, log-canonically polarized pairs over an algebraically closed field of characteristic p >0 such that p ind(K_X/Y + ). We show that K_X/Y + is nef and that f_* (O_X(m (K_X/Y + ))) is a nef vector bundle for m 0 and divisible enough. Some of the results also extend to non log-canonically polarized pairs. The main motivation of the above results is projectivity of proper subspaces of the moduli space of stable pairs in positive characteristics. Other applications are Kodaira vanishing free, algebraic proofs of corresponding positivity results in characteristic zero, and special cases of subadditivity of Kodaira-dimension in positive characteristics.

Keywords: Semi-positive, nef, relative canonical sheaf, sharply F-pure, strongly F-regular, stable varieties, subadditivity of Kodaira dimension.

Class. math. : 14J10, 14J20.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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