Morrison-Kawamata cone conjecture  for hyperkähler manifolds

Amerik, Ekaterina; Verbitsky, Misha

doi:10.24033/asens.2336

Geodesic

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Morrison-Kawamata cone conjecture for hyperkähler manifolds
[Conjecture de Morrison-Kawamata pour les variétés hyperkählériennes]

Amerik, Ekaterina ; Verbitsky, Misha

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 973-993.

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Abstract (VO)
Résumé

Let $M$ be a simple hyperkähler manifold, that is, a simply connected compact holomorphically symplectic manifold of Kähler type with $h^{2, 0} = 1$ . Assuming $b_{2} (M) \neq 5$ , we prove that the group of holomorphic automorphisms of $M$ acts on the set of faces of its Kähler cone with finitely many orbits. This statement is known as Morrison-Kawamata cone conjecture for hyperkähler manifolds. As an implication, we show that a hyperkähler manifold has only finitely many non-equivalent birational models. The proof is based on the following observation, proven with ergodic theory. Let $M$ be a complete Riemannian manifold of dimension at least three, constant negative curvature and finite volume, and ${S_{i}}$ an infinite set of complete, locally geodesic hypersurfaces. Then the union of $S_{i}$ is dense in $M$ .

Soit $M$ une variété hyperkählérienne irréductible. En supposant $b_{2} (M) \neq 5$ , nous montrons que le groupe d'automorphismes de $M$ n'a qu'un nombre fini d'orbites sur l'ensemble des faces du cône de Kähler. Cet enoncé est une version de la conjecture de Morrison-Kawamata pour les variétés hyperkählériennes. Une conséquence en est la finitude du nombre des modèles birationnels pour une telle variété. La preuve s'appuie sur l'observation suivante, qui se démontre dans le cadre de la théorie ergodique : soient $M$ une variété riemanienne complète de dimension au moins trois, de courbure constante négative et de volume fini, et ${S_{i}}$ un ensemble infini d'hypersurfaces localement géodésiques. Alors la réunion des $S_{i}$ est dense dans $M$ .

Publié le : 2017-05-27

MR Zbl

DOI : 10.24033/asens.2336

Keywords: 53C26, 32G13
Mots-clés : Variété hyperkählerienne, espace de modules, application de périodes, théorème de Torelli.

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     title = {Morrison-Kawamata cone conjecture  for hyperk\"ahler manifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {973--993},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
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Amerik, Ekaterina; Verbitsky, Misha. Morrison-Kawamata cone conjecture  for hyperkähler manifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 4, pp. 973-993. doi : 10.24033/asens.2336. http://geodesic.mathdoc.fr/articles/10.24033/asens.2336/

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