Some constraints on positive entropy automorphisms of smooth threefolds

Lesieutre, John

doi:10.24033/asens.2380

Some constraints on positive entropy automorphisms of smooth threefolds
[Quelques contraintes sur les automorphismes d'entropie positive de variétés lisses projectives de dimension trois]

Lesieutre, John

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1507-1547.

Résumé
Abstract

Soit $X$ une variété projective lisse de dimension trois sur $ℂ$ . Nous supposons qu'il existe un automorphisme $ϕ : X ⤏ X$ d'entropie positive. Quitte à remplacer $ϕ$ par un de ses itérés $ϕ^{n}$ , 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 $K_{X}$ -négatifs sur $\bar{N E} (X)$ .

Suppose that $X$ is a smooth, projective threefold over $ℂ$ and that $ϕ : X \to X$ 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 $K_{X}$ -negative extremal rays on $\bar{N E} (X)$ .

MR Zbl

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.

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     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},
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     year = {2018},
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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://archive.numdam.org/articles/10.24033/asens.2380/

Bibliographie
Cité par

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

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