Soit une variété projective lisse de dimension trois sur . Nous supposons qu'il existe un automorphisme d'entropie positive. Quitte à remplacer par un de ses itérés , nous montrons qu'une des affirmations suivantes sera verifiée : i) la classe canonique de est numériquement triviale ; ii) est imprimitive ; iii) n'est pas dynamiquement minimal. Comme corollaire, nous montrons que si une variété lisse 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 .
Notre méthode ne s'applique pas dans le cadre des variétés à singularités terminales. Ceci sera illustré par l'exemple d'une variété uniréglée qui admet une infinité de rayons extrémaux -négatifs sur .
Suppose that is a smooth, projective threefold over and that is an automorphism of positive entropy. We show that one of the following must hold, after replacing by an iterate: i) the canonical class of is numerically trivial; ii) is imprimitive; iii) is not dynamically minimal. As a consequence, we show that if a smooth threefold does not admit a primitive automorphism of positive entropy, then no variety constructed by a sequence of smooth blow-ups of 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 with infinitely many -negative extremal rays on .
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/
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