Some constraints on positive entropy automorphisms of smooth threefolds
[Quelques contraintes sur les automorphismes d'entropie positive de variétés lisses projectives de dimension trois]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1507-1547.

Soit X une variété projective lisse de dimension trois sur . Nous supposons qu'il existe un automorphisme ϕ:XX d'entropie positive. Quitte à remplacer ϕ par un de ses itérés ϕn, nous montrons qu'une des affirmations suivantes sera verifiée : i) la classe canonique de X est numériquement triviale ; ii) ϕ est imprimitive ; iii) ϕ n'est pas dynamiquement minimal. Comme corollaire, nous montrons que si une variété lisse M de dimension trois n'admet pas d'automorphisme primitif d'entropie positive, il en est de même pour toute variété construite par une suite d'éclatements lisses de M.

Notre méthode ne s'applique pas dans le cadre des variétés à singularités terminales. Ceci sera illustré par l'exemple d'une variété uniréglée X qui admet une infinité de rayons extrémaux KX-négatifs sur NE¯(X).

Suppose that X is a smooth, projective threefold over  and that ϕ:XX is an automorphism of positive entropy. We show that one of the following must hold, after replacing ϕ by an iterate: i) the canonical class of X is numerically trivial; ii) ϕ is imprimitive; iii) ϕ is not dynamically minimal. As a consequence, we show that if a smooth threefold M does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of M can admit a primitive automorphism of positive entropy.

In explaining why the method does not apply to threefolds with terminal singularities, we exhibit a non-uniruled, terminal threefold X with infinitely many KX-negative extremal rays on NE¯(X).

DOI : 10.24033/asens.2380
Classification : 14J50, 14E07, 37F99, 14E30.
Keywords: Positive entropy automorphisms, minimal model program, threefolds
Mot clés : Automorphismes d'entropie positive, programme du modèle minimal, variétés.
@article{ASENS_2018__51_6_1507_0,
     author = {Lesieutre, John},
     title = {Some constraints on positive entropy automorphisms of smooth threefolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1507--1547},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {6},
     year = {2018},
     doi = {10.24033/asens.2380},
     mrnumber = {3940903},
     zbl = {1431.14035},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2380/}
}
TY  - JOUR
AU  - Lesieutre, John
TI  - Some constraints on positive entropy automorphisms of smooth threefolds
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2018
SP  - 1507
EP  - 1547
VL  - 51
IS  - 6
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2380/
DO  - 10.24033/asens.2380
LA  - en
ID  - ASENS_2018__51_6_1507_0
ER  - 
%0 Journal Article
%A Lesieutre, John
%T Some constraints on positive entropy automorphisms of smooth threefolds
%J Annales scientifiques de l'École Normale Supérieure
%D 2018
%P 1507-1547
%V 51
%N 6
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2380/
%R 10.24033/asens.2380
%G en
%F ASENS_2018__51_6_1507_0
Lesieutre, John. Some constraints on positive entropy automorphisms of smooth threefolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1507-1547. doi : 10.24033/asens.2380. http://www.numdam.org/articles/10.24033/asens.2380/

Arnol'd, V. I., Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, 1993, pp. 379-390 | MR | Zbl

Bisi, C.; Cascini, P.; Tasin, L. A remark on the Ueno-Campana's threefold, Michigan Math. J., Volume 65 (2016), pp. 567-572 (ISSN: 0026-2285) | DOI | MR | Zbl

Bedford, E.; Diller, J.; Kim, K., Complex geometry and dynamics (Abel Symp.), Volume 10, Springer, 2015, pp. 1-27 | DOI | MR | Zbl

Birkhoff, G. Linear transformations with invariant cones, Amer. Math. Monthly, Volume 74 (1967), pp. 274-276 (ISSN: 0002-9890) | DOI | MR | Zbl

Bedford, E.; Kim, K. Periodicities in linear fractional recurrences: degree growth of birational surface maps, Michigan Math. J., Volume 54 (2006), pp. 647-670 (ISSN: 0026-2285) | DOI | MR | Zbl

Bedford, E.; Kim, K. Dynamics of rational surface automorphisms: linear fractional recurrences, J. Geom. Anal., Volume 19 (2009), pp. 553-583 (ISSN: 1050-6926) | DOI | MR | Zbl

Blanc, J. Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces, Indiana Univ. Math. J., Volume 62 (2013), pp. 1143-1164 (ISSN: 0022-2518) | DOI | MR | Zbl

Cantat, S. Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris Sér. I Math., Volume 328 (1999), pp. 901-906 (ISSN: 0764-4442) | DOI | MR | Zbl

Debarre, O., Universitext, Springer, 2001, 233 pages (ISBN: 0-387-95227-6) | DOI | MR | Zbl

Dinh, T.-C.; Sibony, N. Groupes commutatifs d'automorphismes d'une variété kählérienne compacte, Duke Math. J., Volume 123 (2004), pp. 311-328 (ISSN: 0012-7094) | DOI | MR | Zbl

Eisenbud, D., Graduate Texts in Math., 150, Springer, 1995, 785 pages (ISBN: 0-387-94268-8; 0-387-94269-6) | DOI | MR | Zbl

Hartshorne, R., Graduate Texts in Math., 52, Springer, 1977, 496 pages (ISBN: 0-387-90244-9) | MR | Zbl

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

Kawamata, Y. Abundance theorem for minimal threefolds, Invent. math., Volume 108 (1992), pp. 229-246 (ISSN: 0020-9910) | DOI | MR | Zbl

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

Kawamata, Y.; Matsuda, K.; Matsuki, K., Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, 1987, pp. 283-360 | DOI | MR | Zbl

Kollár, J., Analytic and algebraic geometry (IAS/Park City Math. Ser.), Volume 17, Amer. Math. Soc., 2010, pp. 495-524 | DOI | MR | Zbl

Kollár, J., Ergebn. Math. Grenzg., 32, Springer, 1996, 320 pages (ISBN: 3-540-60168-6) | DOI | MR | Zbl

Lazarsfeld, R., Ergebn. Math. Grenzg., 48, Springer, 2004, 387 pages (ISBN: 3-540-22534-X) | DOI | MR | Zbl

Lesieutre, J.; Litt, D. Dynamical Mordell-Lang and automorphisms of blow-ups (preprint arXiv:1604.08216, to appear in Algebraic Geometry ) | MR

McMullen, C. T. Dynamics on blowups of the projective plane, Publ. Math. Inst. Hautes Études Sci., Volume 105 (2007), pp. 49-89 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Mori, S. Threefolds whose canonical bundles are not numerically effective, Ann. of Math., Volume 116 (1982), pp. 133-176 (ISSN: 0003-486X) | DOI | MR | Zbl

Oguiso, K. Some aspects of explicit birational geometry inspired by complex dynamics, Proceedings of the International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa (2014), pp. 695-721 | MR | Zbl

Oguiso, K.; Truong, T. T. Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy, J. Math. Sci. Univ. Tokyo, Volume 22 (2015), pp. 361-385 (ISSN: 1340-5705) | MR | Zbl

Perroni, F.; Zhang, D.-Q. Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces, Math. Ann., Volume 359 (2014), pp. 189-209 (ISSN: 0025-5831) | DOI | MR | Zbl

Sernesi, E., Grundl. math. Wiss., 334, Springer, 2006, 339 pages (ISBN: 978-3-540-30608-5; 3-540-30608-0) | MR | Zbl

Totaro, B. The topology of smooth divisors and the arithmetic of abelian varieties, Michigan Math. J., Volume 48 (2000), pp. 611-624 (ISSN: 0026-2285) | DOI | MR | Zbl

Totaro, B. Moving codimension-one subvarieties over finite fields, Amer. J. Math., Volume 131 (2009), pp. 1815-1833 (ISSN: 0002-9327) | DOI | MR | Zbl

Truong, T. T. (Relative) dynamical degrees of rational maps over an algebraic closed field (preprint arXiv:1501.01523 )

Truong, T. T. Automorphisms of blowups of threefolds being Fano or having Picard number 1, Ergodic Theory Dynam. Systems, Volume 37 (2017), pp. 2255-2275 (ISSN: 0143-3857) | DOI | MR | Zbl

Uehara, H. Calabi-Yau threefolds with infinitely many divisorial contractions, J. Math. Kyoto Univ., Volume 44 (2004), pp. 99-118 (ISSN: 0023-608X) | DOI | MR | Zbl

Wiśniewski, J. A. On contractions of extremal rays of Fano manifolds, J. reine angew. Math., Volume 417 (1991), pp. 141-157 (ISSN: 0075-4102) | DOI | MR | Zbl

Zhang, D.-Q. Dynamics of automorphisms on projective complex manifolds, J. Differential Geom., Volume 82 (2009), pp. 691-722 http://projecteuclid.org/euclid.jdg/1251122550 (ISSN: 0022-040X) | MR | Zbl

Zhang, D.-Q. n-dimensional projective varieties with the action of an abelian group of rank n-1 , Trans. Amer. Math. Soc., Volume 368 (2016), pp. 8849-8872 (ISSN: 0002-9947) | DOI | MR | Zbl

Cité par Sources :