Holomorphic actions, Kummer examples, and Zimmer program

Cantat, Serge; Zeghib, Abdelghani

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 3, pp. 447-489.

Nous classons les variétés compactes kählériennes $M$ de dimension $n\ge 3$ munies d’une action d’un réseau $\Gamma$ dans un groupe de Lie réel presque simple de rang $\ge n-1$. Ceci complète le programme de Zimmer dans ce cadre, et caractérise certains tores complexes compacts par des propriétés de leur groupe d’automorphismes.

We classify compact Kähler manifolds $M$ of dimension $n\ge 3$ on which acts a lattice of an almost simple real Lie group of rank $\ge n-1$. This provides a new line in the so-called Zimmer program, and characterizes certain complex tori as compact Kähler manifolds with large automorphisms groups.

Classification : 22E40, 32J27
Keywords: lattices, superrigidity, complex tori, automorphism groups, Hodge theory, invariant cones, holomorphic dynamics
Mot clés : réseaux, super-rigidité, tores complexes, groupes d'automorphismes, théorie de Hodge, cônes invariants, dynamique holomorphe

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