Holomorphic actions, Kummer examples, and Zimmer program

[Actions holomorphes, exemples de Kummer et programme de Zimmer]

Cantat, Serge ; Zeghib, Abdelghani

Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012), p. 447-489 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2012-01-01

MR 3014483 | Zbl 1280.22015 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2170
Classification:  22E40,  32J27
Mots clés: réseaux, super-rigidité, tores complexes, groupes d'automorphismes, théorie de Hodge, cônes invariants, dynamique holomorphe

Bibliographie

[1] H. Abels, G. A. Margulis & G. A. Soifer, Semigroups containing proximal linear maps, Israel J. Math. 91 (1995), 1-30. | MR 1348303 | Zbl 0845.22004

[2] D. N. Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, 1995. | MR 1334091 | Zbl 0845.22001

[3] B. Bekka, P. De La Harpe & A. Valette, Kazhdan's property (T), New Mathematical Monographs 11, Cambridge Univ. Press, 2008. | MR 2415834 | Zbl 1146.22009

[4] Y. Benoist, Sous-groupes discrets des groupes de Lie, in European Summer School in Group Theory, 1997, 1-72.

[5] Y. Benoist, Automorphismes des cônes convexes, Invent. Math. 141 (2000), 149-193. | MR 1767272 | Zbl 0957.22008

[6] Y. Benoist, Réseaux des groupes de Lie, cours de Master 2, Université Paris VI, p. 1-72, 2008.

[7] R. Berman & J.-P. Demailly, Regularity of plurisubharmonic upper envelopes in big cohomology classes, in Perspectives in analysis, geometry, and topology, Progr. Math. 296, Birkhäuser, 2012, 39-66. | MR 2884031 | Zbl 1258.32010

[8] A. L. Besse, Einstein manifolds, Classics in Mathematics, Springer, 2008. | MR 2371700 | Zbl 1147.53001

[9] C. Birkenhake & H. Lange, Complex tori, Progress in Math. 177, Birkhäuser, 1999. | MR 1713785 | Zbl 0945.14027

[10] C. Birkenhake & H. Lange, Complex Abelian varieties, 2nd éd., Grund. Math. Wiss. 302, Springer, 2004. | MR 2062673 | Zbl 1056.14063

[11] S. Bochner & D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. 47 (1946), 639-653. | MR 18187 | Zbl 0061.04407

[12] A. Borel, Les bouts des espaces homogènes de groupes de Lie, Ann. of Math. 58 (1953), 443-457. | MR 57263 | Zbl 0053.13002

[13] F. Campana, Orbifoldes à première classe de Chern nulle, in The Fano Conference, Univ. Torino, Turin, 2004, 339-351. | Zbl 1068.53051

[14] F. Campana & T. Peternell, Cycle spaces, in Several complex variables, VII, Encyclopaedia Math. Sci. 74, Springer, 1994, 319-349. | Zbl 0811.32020

[15] S. Cantat, Sur la dynamique du groupe d’automorphismes des surfaces $K3$, Transform. Groups 6 (2001), 201-214. | Zbl 1081.14513

[16] S. Cantat, Version kählérienne d'une conjecture de Robert J. Zimmer, Ann. Sci. École Norm. Sup. 37 (2004), 759-768. | Numdam | Zbl 1072.22006

[17] S. Cantat, Caractérisation des exemples de Lattès et de Kummer, Compos. Math. 144 (2008), 1235-1270. | Zbl 1156.37012

[18] S. Cantat, Sur les groupes de transformations birationnelles des surfaces, Ann. of Math. 174 (2011), 299-340. | Zbl 1233.14011

[19] S. Cantat & A. Zeghib, Holomorphic actions of higer rank lattices in dimension three, preprint, 2009.

[20] S. Cantat & A. Zeghib, Holomorphic actions, Kummer examples, and Zimmer program, preprint, 2010. | Numdam | Zbl 1280.22015

[21] J.-P. Demailly, Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France (N.S.) 19 (1985). | Numdam | Zbl 0579.32012

[22] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), 361-409. | Zbl 0777.32016

[23] J.-P. Demailly & M. Păun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. 159 (2004), 1247-1274. | Zbl 1064.32019

[24] T.-C. Dinh & V.-A. Nguyên, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv. 86 (2011), 817-840. | MR 2851870 | Zbl 1279.32018

[25] T.-C. Dinh & N. Sibony, Groupes commutatifs d'automorphismes d'une variété kählérienne compacte, Duke Math. J. 123 (2004), 311-328. | MR 2066940 | Zbl 1065.32012

[26] I. V. Dolgachev & D.-Q. Zhang, Coble rational surfaces, Amer. J. Math. 123 (2001), 79-114. | MR 1827278 | Zbl 1056.14054

[27] D. Fisher, Groups acting on manifolds: around the Zimmer program, in Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, 2011, 72-157. | MR 2807830 | Zbl 1264.22012

[28] A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), 225-258. | MR 481142 | Zbl 0367.32004

[29] W. Fulton, Introduction to toric varieties, Annals of Math. Studies 131, Princeton Univ. Press, 1993. | MR 1234037 | Zbl 0813.14039

[30] W. Fulton & J. Harris, Representation theory, Graduate Texts in Math. 129, Springer, 1991. | MR 1153249 | Zbl 0744.22001

[31] É. Ghys, Actions de réseaux sur le cercle, Invent. Math. 137 (1999), 199-231. | MR 1703323 | Zbl 0995.57006

[32] B. Gilligan & A. T. Huckleberry, Complex homogeneous manifolds with two ends, Michigan Math. J. 28 (1981), 183-198. | MR 616269 | Zbl 0452.32022

[33] V. V. Gorbatsevich, O. V. Shvartsman & È. B. Vinberg (éds.), Discrete subgroups of Lie groups, in Lie groups and Lie algebras. II, Encyclopaedia of Math. Sciences 21, Springer, 2000. | MR 1756406 | Zbl 0931.22007

[34] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. | MR 137127 | Zbl 0173.33004

[35] P. Griffiths & J. Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons Inc., 1994. | MR 1288523 | Zbl 0836.14001

[36] M. Gromov, On the entropy of holomorphic maps, Enseign. Math. 49 (2003), 217-235. | MR 2026895 | Zbl 1080.37051

[37] E. Guentner, N. Higson & S. Weinberger, The Novikov conjecture for linear groups, Publ. Math. IHÉS 101 (2005), 243-268. | Numdam | MR 2217050 | Zbl 1073.19003

[38] P. De La Harpe & A. Valette, La propriété $(T)$ de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger), Astérisque 175 (1989). | Zbl 0759.22001

[39] R. Hartshorne, Ample subvarieties of algebraic varieties, Notes written in collaboration with C. Musili. Lecture Notes in Math. 156, Springer, 1970. | MR 282977 | Zbl 0208.48901

[40] A. T. Huckleberry & D. M. Snow, Almost-homogeneous Kähler manifolds with hypersurface orbits, Osaka J. Math. 19 (1982), 763-786. | MR 687772 | Zbl 0507.32023

[41] S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan 15, Princeton Univ. Press, 1987. | MR 909698 | Zbl 0708.53002

[42] K. Kodaira & D. C. Spencer, A theorem of completeness of characteristic systems of complete continuous systems, Amer. J. Math. 81 (1959), 477-500. | MR 112156 | Zbl 0097.36501

[43] A. G. Kušnirenko, An analytic action of a semisimple Lie group in a neighborhood of a fixed point is equivalent to a linear one, Funkcional. Anal. i Priložen 1 (1967), 103-104. | MR 210833 | Zbl 0156.42205

[44] S. Lang, Algebra, second éd., Addison-Wesley Publishing Company Advanced Book Program, 1984. | MR 783636 | Zbl 0712.00001

[45] R. Lazarsfeld, Positivity in algebraic geometry. I and II, Ergebn. Math. Grenzg. 48/49, Springer, 2004. | MR 2095471 | Zbl 1093.14500

[46] D. I. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, in Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975-1977), Lecture Notes in Math. 670, Springer, 1978, 140-186. | MR 521918 | Zbl 0391.32018

[47] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebn. Math. Grenzg. (3) 17, Springer, 1991. | MR 1090825 | Zbl 0732.22008

[48] G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies 78, Princeton Univ. Press, 1973. | MR 385004 | Zbl 0265.53039

[49] M. Nakamaye, Base loci of linear series are numerically determined, Trans. Amer. Math. Soc. 355 (2003), 551-566 (electronic). | MR 1932713 | Zbl 1017.14017

[50] A. L. Onishchik & È. B. Vinberg (éds.), Lie groups and Lie algebras, III, Encyclopaedia of Math. Sciences 41, Springer, 1994. | MR 1349140 | Zbl 0797.22001

[51] G. Prasad & M. S. Raghunathan, Cartan subgroups and lattices in semi-simple groups, Ann. of Math. 96 (1972), 296-317. | MR 302822 | Zbl 0245.22013

[52] W. M. Ruppert, Two-dimensional complex tori with multiplication by $\sqrt{d}$, Arch. Math. (Basel) 72 (1999), 278-281. | MR 1678033 | Zbl 0964.14038

[53] I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359-363. | Zbl 0074.18103

[54] A. Shimizu, On complex tori with many endomorphisms, Tsukuba J. Math. 8 (1984), 297-318. | Zbl 0581.14032

[55] J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, in Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, 525-563. | Zbl 0373.14007

[56] D. Sullivan, Infinitesimal computations in topology, Publ. Math. I.H.É.S. 47 (1977), 269-331 (1978). | Numdam | Zbl 0374.57002

[57] È. B. Vinberg & V. G. Kac, Quasi-homogeneous cones, Mat. Zametki 1 (1967), 347-354. | Zbl 0163.16902

[58] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Soc. Math. France, 2002. | Zbl 1032.14001

[59] Z. Z. Wang & D. Zaffran, A remark on the hard Lefschetz theorem for Kähler orbifolds, Proc. Amer. Math. Soc. 137 (2009), 2497-2501. | Zbl 1169.14015

[60] J. A. Wolf, Spaces of constant curvature, fifth éd., Publish or Perish Inc., 1984. | Zbl 0281.53034

[61] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), 339-411. | Zbl 0369.53059

[62] D.-Q. Zhang, A theorem of Tits type for compact Kähler manifolds, Invent. Math. 176 (2009), 449-459. | Zbl 1170.14029

[63] R. J. Zimmer, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), 425-436. | MR 735334 | Zbl 0576.22014

[64] R. J. Zimmer, Actions of semisimple groups and discrete subgroups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., 1987, 1247-1258. | MR 934329 | Zbl 0671.57028

[65] R. J. Zimmer, Lattices in semisimple groups and invariant geometric structures on compact manifolds, in Discrete groups in geometry and analysis (New Haven, Conn., 1984), Progr. Math. 67, Birkhäuser, 1987, 152-210. | MR 900826 | Zbl 0663.22008