Semi-positivity in positive characteristics

Patakfalvi, Zsolt

doi:10.24033/asens.2232

Semi-positivity in positive characteristics
[Semi-positivité en caractéristiques positives]

Patakfalvi, Zsolt

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 5, pp. 991-1025.

Résumé
Abstract

Soit $f : (X, Δ) \to Y$ une famille projective plate de paires nettement $F$ -pures et log-canoniquement polarisées sur un corps algébriquement clos de caractéristique $p > 0$ tel que $p ∤ ind (K_{X / Y} + Δ)$ . Nous montrons que $K_{X / Y} + Δ$ est nef et que $f_{*} (𝒪_{X} (m (K_{X / Y} + Δ)))$ 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.

Let $f : (X, Δ) \to Y$ be a flat, projective family of sharply $F$ -pure, log-canonically polarized pairs over an algebraically closed field of characteristic $p > 0$ such that $p ∤ ind (K_{X / Y} + Δ)$ . We show that $K_{X / Y} + Δ$ is nef and that $f_{*} (𝒪_{X} (m (K_{X / Y} + Δ)))$ 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.

Publié le : 2014-09-24

MR Zbl

DOI : 10.24033/asens.2232

Classification : 14J10, 14J20.
Keywords: Semi-positive, nef, relative canonical sheaf, sharply $F$-pure, strongly $F$-regular, stable varieties, subadditivity of Kodaira dimension.
Mot clés : Semi-positif, nef, faisceau canonique relatif, nettement $F$-pur, fortement $F$-régulier, variétés stables, sous-additivité de la dimension de Kodaira.

@article{ASENS_2014__47_5_991_0,
     author = {Patakfalvi, Zsolt},
     title = {Semi-positivity in positive characteristics},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {991--1025},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 47},
     number = {5},
     year = {2014},
     doi = {10.24033/asens.2232},
     mrnumber = {3294622},
     zbl = {1326.14015},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2232/}
}

TY  - JOUR
AU  - Patakfalvi, Zsolt
TI  - Semi-positivity in positive characteristics
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2014
SP  - 991
EP  - 1025
VL  - 47
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2232/
DO  - 10.24033/asens.2232
LA  - en
ID  - ASENS_2014__47_5_991_0
ER  -

%0 Journal Article
%A Patakfalvi, Zsolt
%T Semi-positivity in positive characteristics
%J Annales scientifiques de l'École Normale Supérieure
%D 2014
%P 991-1025
%V 47
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2232/
%R 10.24033/asens.2232
%G en
%F ASENS_2014__47_5_991_0

Patakfalvi, Zsolt. Semi-positivity in positive characteristics. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 47 (2014) no. 5, pp. 991-1025. doi : 10.24033/asens.2232. http://www.numdam.org/articles/10.24033/asens.2232/

Bibliographie
Cité par

Arakelov, S. J. Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat., Volume 35 (1971), pp. 1269-1293 (ISSN: 0373-2436) | MR | Zbl

Bhatt, B.; Ho, W.; Patakfalvi, Zs.; Schnell, C. Moduli of products of stable varieties, Compositio Math., Volume 149 (2013), pp. 2036-2070 http://journals.cambridge.org/article_S0010437X13007288 (ISSN: 1570-5846) | DOI | MR | Zbl

Cartan, H., Séminaire Henri Cartan; 10e année : 1957/1958. Fonctions automorphes, Secrétariat mathématique, 1958 | MR | Zbl

Eisenbud, D.; Harris, J. The Kodaira dimension of the moduli space of curves of genus $\geq 23$ , Invent. Math., Volume 90 (1987), pp. 359-387 (ISSN: 0020-9910) | DOI | MR | Zbl

Faltings, G.; Chai, C.-L., Ergebn. Math. Grenzg., 22, Springer, 1990, 316 pages (ISBN: 3-540-52015-5) | MR | Zbl

Fujino, O.; Fujisawa, T. Variations of Mixed Hodge Structure and Semipositivity Theorems, Publ. Res. Inst. Math. Sci., Volume 50 (2014), pp. 589-661 (ISSN: 0034-5318) | DOI | MR | Zbl

Fujino, O. Semi-positivity theorems for moduli problems (2012) (preprint) | MR

Fujino, O. Fundamental theorems for semi log canonical pairs, Algebr. Geom., Volume 1 (2014), pp. 194-228 (ISSN: 2214-2584) | DOI | MR | Zbl

Fujita, T. On Kähler fiber spaces over curves, J. Math. Soc. Japan, Volume 30 (1978), pp. 779-794 (ISSN: 0025-5645) | DOI | MR | Zbl

Gabber, O., Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 711-734 | DOI | MR | Zbl

Griffiths, P. A. Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math. I.H.É.S., Volume 38 (1970), pp. 125-180 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math. I.H.É.S., Volume 24 (1965), pp. 5-231 (ISSN: 0073-8301) | Numdam | MR | Zbl

Hartshorne, R., Graduate Texts in Math., 52, Springer, 1977 (ISBN: 0-387-90244-9) | MR | Zbl

Hartshorne, R. Generalized divisors on Gorenstein schemes, $K$ -Theory (Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992)), Volume 8 (1994), pp. 287-339 (ISSN: 0920-3036) | DOI | MR | Zbl

Hara, N. A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math., Volume 120 (1998), pp. 981-996 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5hara.pdf (ISSN: 0002-9327) | DOI | MR | Zbl

Hara, N. Classification of two-dimensional $F$ -regular and $F$ -pure singularities, Adv. Math., Volume 133 (1998), pp. 33-53 (ISSN: 0001-8708) | DOI | MR | Zbl

Hassett, B.; Kovács, S. J. Reflexive pull-backs and base extension, J. Algebraic Geom., Volume 13 (2004), pp. 233-247 (ISSN: 1056-3911) | DOI | MR | Zbl

Hara, N.; Watanabe, K.-I. F-regular and F-pure rings vs. log terminal and log canonical singularities, J. Algebraic Geom., Volume 11 (2002), pp. 363-392 (ISSN: 1056-3911) | DOI | MR | Zbl

Kawamata, Y. Characterization of abelian varieties, Compositio Math., Volume 43 (1981), pp. 253-276 (ISSN: 0010-437X) | Numdam | MR | Zbl

Keeler, D. S. Ample filters of invertible sheaves, J. Algebra, Volume 259 (2003), pp. 243-283 (ISSN: 0021-8693) | DOI | MR | Zbl

Kollár, J.; Mori, S., Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, 1998, 254 pages (ISBN: 0-521-63277-3) | MR | Zbl

Kollár, J. Moduli of varieties of general type, Handbook of moduli, vol. II (Adv. Lect. Math.), Volume 25, Int. Press (2013), pp. 131-157 | MR | Zbl

Kollár, J., Cambridge Tracts in Mathematics, 200, 2013 | MR | Zbl

Kollár, J., Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, 1987, pp. 361-398 | DOI | MR | Zbl

Kollár, J. Projectivity of complete moduli, J. Differential Geom., Volume 32 (1990), pp. 235-268 http://projecteuclid.org/getRecord?id=euclid.jdg/1214445046 (ISSN: 0022-040X) | MR | Zbl

Lazarsfeld, R., Ergebn. Math. Grenzg., 48, Springer, 2004, 387 pages (ISBN: 3-540-22533-1) | MR | Zbl

Maulik, D. Supersingular K3 surfaces for large primes, Duke Math. J., Volume 163 (2014), pp. 2357-2425 | DOI | MR | Zbl

Moret-Bailly, L. Familles de courbes et de variétés abéliennes sur $ℙ^{1}$ (II. Exemples), Astérisque, Volume 86 (1981), pp. 125-140 | MR | Zbl

Mustaţă, M.; Srinivas, V. Ordinary varieties and the comparison between multiplier ideals and test ideals, Nagoya Math. J., Volume 204 (2011), pp. 125-157 http://projecteuclid.org/getRecord?id=euclid.nmj/1323107839 (ISSN: 0027-7630) | DOI | MR | Zbl

Miller, L. E.; Schwede, K. Semi-log canonical vs $F$ -pure singularities, J. Algebra, Volume 349 (2012), pp. 150-164 (ISSN: 0021-8693) | DOI | MR | Zbl

Mehta, V. B.; Srinivas, V. Normal $F$ -pure surface singularities, J. Algebra, Volume 143 (1991), pp. 130-143 (ISSN: 0021-8693) | DOI | MR | Zbl

Mumford, D. Hirzebruch's proportionality theorem in the noncompact case, Invent. Math., Volume 42 (1977), pp. 239-272 (ISSN: 0020-9910) | DOI | MR | Zbl

Mustaţă, M. Ordinary varieties and the comparison between multiplier ideals and test ideals II, Proc. Amer. Math. Soc., Volume 140 (2012), pp. 805-810 (ISSN: 0002-9939) | DOI | MR | Zbl

Paršin, A. N. Algebraic curves over function fields, Dokl. Akad. Nauk SSSR, Volume 183 (1968), pp. 524-526 (ISSN: 0002-3264) | MR | Zbl

Patakfalvi, Zs. Fibered stable varieties (to appear in Trans. of the A.M.S ) | MR

Patakfalvi, Zs. On subadditivity of Kodaira dimension in positive characteristic (preprint arXiv:1308.5371 )

Patakfalvi, Zs.; Schwede, K. Depth of $F$ -singularities and base change of relative canonical sheaves, Journal of the Institute of Mathematics of Jussieu, Volume 13 (2014), pp. 43-63 http://journals.cambridge.org/article_S1474748013000066 (ISSN: 1475-3030) | DOI | MR | Zbl

Patakfalvi, Zs.; Schwede, K.; Zhang, W. $F$ -singularities in families (preprint arXiv:1305.1646 ) | MR

Raynaud, M., C. P. Ramanujam—a tribute (Tata Inst. Fund. Res. Studies in Math.), Volume 8, Springer, 1978, pp. 273-278 | MR | Zbl

Schwede, K. A canonical linear system associated to adjoint divisors in characteristic $p > 0$ (preprint arXiv:1107.3833, to appear in J. reine angew. Math ) | MR | Zbl

Schwede, K. $F$ -adjunction, Algebra Number Theory, Volume 3 (2009), pp. 907-950 (ISSN: 1937-0652) | DOI | MR | Zbl

Schwede, K.; Tucker, K., Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012, pp. 39-99 | MR | Zbl

Szpiro, L., Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. II (Astérisque), Volume 64, Soc. Math. France, 1979, pp. 169-202 | Numdam | MR | Zbl

Takagi, S. An interpretation of multiplier ideals via tight closure, J. Algebraic Geom., Volume 13 (2004), pp. 393-415 (ISSN: 1056-3911) | DOI | MR | Zbl

Viehweg, E., Algebraic varieties and analytic varieties (Tokyo, 1981) (Adv. Stud. Pure Math.), Volume 1, North-Holland, 1983, pp. 329-353 | DOI | MR | Zbl

Viehweg, E., Ergebn. Math. Grenzg., 30, Springer, 1995, 320 pages (ISBN: 3-540-59255-5) | MR | Zbl

Cité par Sources :