Some constraints on positive entropy automorphisms of smooth threefolds

Lesieutre, John

doi:10.24033/asens.2380

Geodesic

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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.

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Abstract (VO)
Résumé

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 $\bar{N E} (X)$ .

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 $\bar{N E} (X)$ .

MR Zbl

DOI : 10.24033/asens.2380

Classification : 14J50, 14E07, 37F99, 14E30.
Keywords: Positive entropy automorphisms, minimal model program, threefolds
Mots-clés : Automorphismes d'entropie positive, programme du modèle minimal, variétés.

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     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},
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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://geodesic.mathdoc.fr/articles/10.24033/asens.2380/

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