Albanese maps of projective manifolds with nef anticanonical bundles

Cao, Junyan

doi:10.24033/asens.2405

Albanese maps of projective manifolds with nef anticanonical bundles
[Application d'Albanese de variétés projectives à fibrés anticanoniques nef]

Cao, Junyan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1137-1154.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé
Abstract

Soit $X$ une variété projective à fibré anticanonique nef. On montre que l'application d'Albanese $p : X \to Y$ est localement triviale. En particulier, $p$ est lisse.

Let $X$ be a projective manifold such that the anticanonical bundle $- K_{X}$ is nef. We prove that the Albanese map $p : X \to Y$ is locally trivial. In particular, $p$ is a submersion.

Publié le : 2019-11-12

MR Zbl

DOI : 10.24033/asens.2405

Classification : 32J27; 32Qxx.
Keywords: Nef anticanonical bundle, Albanese map, positivity of direct images.
Mot clés : Fibré anticanonique nef, application d'Albanese, positivité de l'image directe

@article{ASENS_2019__52_5_1137_0,
     author = {Cao, Junyan},
     title = {Albanese maps of projective manifolds with nef anticanonical bundles},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1137--1154},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {5},
     year = {2019},
     doi = {10.24033/asens.2405},
     mrnumber = {4057779},
     zbl = {1442.32026},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2405/}
}

TY  - JOUR
AU  - Cao, Junyan
TI  - Albanese maps of projective manifolds with nef anticanonical bundles
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2019
SP  - 1137
EP  - 1154
VL  - 52
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2405/
DO  - 10.24033/asens.2405
LA  - en
ID  - ASENS_2019__52_5_1137_0
ER  -

%0 Journal Article
%A Cao, Junyan
%T Albanese maps of projective manifolds with nef anticanonical bundles
%J Annales scientifiques de l'École Normale Supérieure
%D 2019
%P 1137-1154
%V 52
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2405/
%R 10.24033/asens.2405
%G en
%F ASENS_2019__52_5_1137_0

Cao, Junyan. Albanese maps of projective manifolds with nef anticanonical bundles. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1137-1154. doi : 10.24033/asens.2405. http://www.numdam.org/articles/10.24033/asens.2405/

Bibliographie
Cité par

Birkar, C.; Chen, Y. Images of manifolds with semi-ample anti-canonical divisor, J. Algebraic Geom., Volume 25 (2016), pp. 273-287 (ISSN: 1056-3911) | DOI | MR | Zbl

Berndtsson, B.; Paun, M. Bergman kernels and subadjunction (preprint arXiv:1002.4145 )

Bauer, T.; Peternell, T. Nef reduction and anticanonical bundles, Asian J. Math., Volume 8 (2004), pp. 315-352 http://projecteuclid.org/euclid.ajm/1088090064 (ISSN: 1093-6106) | DOI | MR | Zbl

Berndtsson, B.; Paun, M. Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J., Volume 145 (2008), pp. 341-378 (ISSN: 0012-7094) | DOI | MR | Zbl

Campana, F. Remarques sur les groupes de Kähler nilpotents, Ann. Sci. École Norm. Sup., Volume 28 (1995), pp. 307-316 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Cao, J. Théorèmes d'annulation et théorèmes de structure sur les variétés kähleriennes compactes (2013)

Campana, F.; Demailly, J.-P.; Peternell, T., Recent advances in algebraic geometry (London Math. Soc. Lecture Note Ser.), Volume 417, Cambridge Univ. Press, 2015, pp. 71-91 | DOI | MR | Zbl

Cao, J.; Höring, A. Manifolds with nef anticanonical bundle, J. reine angew. Math., Volume 724 (2017), pp. 203-244 (ISSN: 0075-4102) | DOI | MR | Zbl

Cao, J.; Höring, A. A decomposition theorem for projective manifolds with nef anticanonical bundle, J. Algebraic Geom., Volume 28 (2019), pp. 567-597 (ISSN: 1056-3911) | DOI | MR | Zbl

Cao, J.; Paun, M. Kodaira dimension of algebraic fiber spaces over abelian varieties, Invent. math., Volume 207 (2017), pp. 345-387 (ISSN: 0020-9910) | DOI | MR | Zbl

Campana, F.; Peternell, T. Projective manifolds whose tangent bundles are numerically effective, Math. Ann., Volume 289 (1991), pp. 169-187 (ISSN: 0025-5831) | DOI | MR | Zbl

Campana, F.; Peternell, T.; Zhang, Q. On the Albanese maps of compact Kähler manifolds, Proc. Amer. Math. Soc., Volume 131 (2003), pp. 549-553 (ISSN: 0002-9939) | DOI | MR | Zbl

Chen, M.; Zhang, Q. On a question of Demailly-Peternell-Schneider, J. Eur. Math. Soc. (JEMS), Volume 15 (2013), pp. 1853-1858 (ISSN: 1435-9855) | DOI | MR | Zbl

Demailly, J.-P., Surveys of Modern Mathematics, 1, International Press; Higher Education Press, 2012, 231 pages (ISBN: 978-1-57146-234-3) | MR | Zbl

Deng, Y. Applications of the Ohsawa-Takegoshi Extension Theorem to Direct Image Problems (preprint arXiv:1703.07279 ) | MR

Deng, Y. Generalized Okounkov body, hyperbolicity and direct image problems (2018)

Demailly, J.-P.; Peternell, T.; Schneider, M. Kähler manifolds with numerically effective Ricci class, Compos. math., Volume 89 (1993), pp. 217-240 (ISSN: 0010-437X) | Numdam | MR | Zbl

Demailly, J.-P.; Peternell, T.; Schneider, M. Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom., Volume 3 (1994), pp. 295-345 (ISSN: 1056-3911) | MR | Zbl

Demailly, J.-P.; Peternell, T.; Schneider, M. Compact Kähler manifolds with Hermitian semipositive anticanonical bundle, Compos. math., Volume 101 (1996), pp. 217-224 (ISSN: 0010-437X) | Numdam | MR | Zbl

Fujino, O.; Gongyo, Y. On images of weak Fano manifolds, Math. Z., Volume 270 (2012), pp. 531-544 (ISSN: 0025-5874) | DOI | MR | Zbl

Fujino, O. Direct images of relative pluricanonical bundles, Algebr. Geom., Volume 3 (2016), pp. 50-62 (ISSN: 2214-2584) | DOI | MR | Zbl

Höring, A. Positivity of direct image sheaves—a geometric point of view, Enseign. Math., Volume 56 (2010), pp. 87-142 (ISSN: 0013-8584) | DOI | MR | Zbl

Kawamata, Y. Kodaira dimension of algebraic fiber spaces over curves, Invent. math., Volume 66 (1982), pp. 57-71 (ISSN: 0020-9910) | DOI | MR | Zbl

Kawamata, Y. Subadjunction of log canonical divisors. II, Amer. J. Math., Volume 120 (1998), pp. 893-899 http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5kawamata.pdf (ISSN: 0002-9327) | DOI | MR | Zbl

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

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

Lu, S.; Tu, Y.; Zhang, Q.; Zheng, Q. On semistability of Albanese maps, Manuscripta Math., Volume 131 (2010), pp. 531-535 (ISSN: 0025-2611) | DOI | MR | Zbl

Manivel, L. Un théorème de prolongement $L^{2}$ de sections holomorphes d'un fibré hermitien, Math. Z., Volume 212 (1993), pp. 107-122 (ISSN: 0025-5874) | DOI | MR | Zbl

Mumford, D.; Fogarty, J.; Kirwan, F., Ergebn. Math. Grenzg., 34, Springer, 1994, 292 pages (ISBN: 3-540-56963-4) | DOI | MR

Ou, W. On generic nefness of tangent sheaves (preprint arXiv:1703.03175 ) | MR

Paun, M., Higher dimensional algebraic geometry—in honour of Professor Yujiro Kawamata's sixtieth birthday (Adv. Stud. Pure Math.), Volume 74, Math. Soc. Japan, Tokyo, 2017, pp. 335-356 | DOI | MR | Zbl

Paun, M. Sur le groupe fondamental des variétés kählériennes compactes à classe de Ricci numériquement effective, C. R. Acad. Sci. Paris Sér. I Math., Volume 324 (1997), pp. 1249-1254 (ISSN: 0764-4442) | DOI | MR | Zbl

Peternell, T.; Serrano, F. Threefolds with nef anticanonical bundles, Collect. Math., Volume 49 (1998), pp. 465-517 (ISSN: 0010-0757) | MR | Zbl

Paun, M.; Takayama, S. Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebraic Geom., Volume 27 (2018), pp. 211-272 (ISSN: 1056-3911) | DOI | MR | Zbl

Simpson, C. T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., Volume 75 (1992), pp. 5-95 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Tsuji, H. Global generation of the direct images of relative pluricanonical systems (preprint arXiv:1012.0884 )

Verbitsky, M., Algebraic structures and moduli spaces (CRM Proc. Lecture Notes), Volume 38, Amer. Math. Soc., 2004, pp. 229-247 | 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) | DOI | MR | Zbl

Zhang, Q. On projective varieties with nef anticanonical divisors, Math. Ann., Volume 332 (2005), pp. 697-703 (ISSN: 0025-5831) | DOI | MR | Zbl

Zhang, Q. On projective manifolds with nef anticanonical bundles, J. reine angew. Math., Volume 478 (1996), pp. 57-60 (ISSN: 0075-4102) | DOI | MR | Zbl

Cité par Sources :