×

zbMATH — the first resource for mathematics

A Kählerian version of a conjecture of Robert J. Zimmer. (Version Kählérienne d’une conjecture de Robert J. Zimmer.) (French) Zbl 1072.22006
We recall the Zimmer’s conjecture. Let \(G\) be a simple connected Lie group and \(\Gamma\) be a lattice in \(G\). Let \(M\) be a compact manifold with group of automorphisms \(\operatorname{Aut}(M)\). The conjecture asserts that if there exists a morphism \(\Gamma \to \operatorname{Aut}(M)\) with infinite image, then the real rank \(r(G)\) of \(G\) is bounded above by the dimension of \(M\). This article is devoted to the proof of this conjecture when \(M\) is a compact complex Kähler manifold: \(r(G)\) is then bounded by the complex dimension of \(M\). The proof consists in reducing the problem to the case where the morphism \(\Gamma \to \operatorname{Aut}(M)\) takes its value in \(\operatorname{Aut}(M)^0\), the connected component of the identity in \(\operatorname{Aut}(M)\). The conjecture then follows by classical arguments. This reduction is done by associating a theorem of Lieberman of Kählerian geometry and a cohomological version of the Zimmer’s conjecture: if the action of the lattice \(\Gamma\) induced on the cohomology of \(M\) has infinite image, then \(r(G)\) is bounded by the complex dimension of \(M\). The proof of this cohomological version is based on results of Prasad and Raghunathan, and a paper by Margulis and on a recent theorem of T.-C. Dinh and N. Sibony [Duke Math. J. 123, No. 2, 311–328 (2004; Zbl 1065.32012)] concerning the rank of the abelian group of automorphisms with positive entropy acting on Kähler manifolds.

MSC:
22E40 Discrete subgroups of Lie groups
32M05 Complex Lie groups, group actions on complex spaces
32J27 Compact Kähler manifolds: generalizations, classification
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Barth W. , Peters C. , Automorphisms of Enriques surfaces , Invent. Math. 73 ( 3 ) ( 1983 ) 383 - 411 . MR 718937 | Zbl 0518.14023 · Zbl 0518.14023
[2] Beauville A. , Variétés kählériennes dont la première classe de Chern est nulle , J. Differential Geom. 18 ( 4 ) ( 1984 ) 755 - 782 , 1983. MR 730926 | Zbl 0537.53056 · Zbl 0537.53056
[3] Beauville A. , Riemannian holonomy and algebric geometry , preprint, math.AG/9902110 , pp. 1-27. arXiv
[4] Benoist Y. , Sous-groupes discrets des groupes de Lie , in: European Summer School in Group Theory , 1997 , pp. 1 - 72 .
[5] Bochner S. , Montgomery D. , Locally compact groups of differentiable transformations , Ann. of Math. (2) 47 ( 1946 ) 639 - 653 . MR 18187 | Zbl 0061.04407 · Zbl 0061.04407
[6] Cairns G. , Ghys É. , The local linearization problem for smooth \(\mathrm{SL}\left(n\right)\)-actions , Enseign. Math. (2) 43 ( 1-2 ) ( 1997 ) 133 - 171 . MR 1460126 | Zbl 0914.57027 · Zbl 0914.57027
[7] Cantat S. , Dynamique des automorphismes des surfaces K 3 , Acta Math. 187 ( 1 ) ( 2001 ) 1 - 57 . MR 1864630 | Zbl 1045.37007 · Zbl 1045.37007
[8] de la Harpe P. , Valette A. , La propriété ( T ) de Kazhdan pour les groupes localement compacts avec un appendice de Marc Burger , Astérisque 175 ( 1989 ) 158 , With an appendix by M. Burger. MR 1023471 | Zbl 0759.22001 · Zbl 0759.22001
[9] Dinh T.-C. , Sibony N. , Groupes commutatifs d’automorphismes d’une variété kählérienne compacte , Duke Math. J. 123 ( 2004 ) 311 - 328 . Article | MR 2066940 | Zbl 1065.32012 · Zbl 1065.32012
[10] Ghys É. , Actions de réseaux sur le cercle , Invent. Math. 137 ( 1 ) ( 1999 ) 199 - 231 . MR 1703323 | Zbl 0995.57006 · Zbl 0995.57006
[11] Gizatullin M.H. , Rational G -surfaces , Izv. Akad. Nauk SSSR Ser. Mat. 44 ( 1 ) ( 1980 ) 110 - 144 , 239. MR 563788 | Zbl 0428.14022 · Zbl 0428.14022
[12] Gromov M. , Convex sets and Kähler manifolds , in: Advances in Differential Geometry and Topology , World Sci. Publishing , Teaneck, NJ , 1990 , pp. 1 - 38 . MR 1095529 | Zbl 0770.53042 · Zbl 0770.53042
[13] Hindry M. , Silverman J.H. , Diophantine Geometry , Graduate Texts in Mathematics , vol. 201 , Springer-Verlag , New York , 2000 , An introduction. MR 1745599 | Zbl 0948.11023 · Zbl 0948.11023
[14] Lieberman D.I. , 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) , Springer , Berlin , 1978 , pp. 140 - 186 . MR 521918 | Zbl 0391.32018 · Zbl 0391.32018
[15] Margulis G.A. , Discrete Subgroups of Semisimple Lie Groups , in: Ergebnisse der Mathematik und ihrer Grenzgebiete (3) , vol. 17 , Springer-Verlag, Berlin , 1991 . MR 1090825 | Zbl 0732.22008 · Zbl 0732.22008
[16] Prasad G. , Raghunathan M.S. , Cartan subgroups and lattices in semi-simple groups , Ann. of Math. 2 96 ( 1972 ) 296 - 317 . MR 302822 | Zbl 0245.22013 · Zbl 0245.22013
[17] Vinberg E.B. , Gorbatsevich V.V. , Shvartsman O.V. , Discrete subgroups of Lie groups , in: Lie Groups and Lie Algebras, II , Encyclopaedia Math. Sci. , vol. 21 , Springer , Berlin , 2000 , pp. 1 - 123 , 217-223. [MR 90c:22036]. MR 1756407 | Zbl 0931.22007 · Zbl 0931.22007
[18] Zimmer R.J. , Actions of semisimple groups and discrete subgroups , in: Proceedings of the International Congress of Mathematicians, vols. 1, 2, (Berkeley, CA, 1986) , Amer. Math. Soc. , Providence, RI , 1987 , pp. 1247 - 1258 . MR 934329 | Zbl 0671.57028 · Zbl 0671.57028
[19] Zimmer R.J. , Lattices in semisimple groups and invariant geometric structures on compact manifolds , in: Discrete Groups in Geometry and Analysis (New Haven, CT, 1984) , Progr. Math. , vol. 67 , Birkhäuser Boston , Boston, MA , 1987 , pp. 152 - 210 . MR 900826 | Zbl 0663.22008 · Zbl 0663.22008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.