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Commutative automorphism groups of a compact Kähler manifold. (Groupes commutatifs d’automorphismes d’une variété kählérienne compacte.) (French. English summary) Zbl 1065.32012
Let \(V\) be a compact Kähler manifold of dimension \(n \geq 2\).
In this paper the notion of positive entropy is used to describe abelian subgroups of \(\text{Aut}(V)\), where by results of M. Gromov [Enseign. Math., II. Sér. 49, No. 3–4, 217–235 (2003; Zbl 1080.37051), Sémin. Bourbaki, 38ème année, Vol. 1985/86, Exposé 663, Astérisque 145/146, 225–240 (1987; Zbl 0611.58041)] and Y. Yomdin [Isr. J. Math. 57, 285–300 (1987; Zbl 0641.54036)] a holomorphic endomorphism of \(V\) has positive entropy if and only if the spectral radius of the induced map \(f^*\) on \({\mathcal H}^{1,1}(V,{\mathbb R})\) is strictly greater than one.
The main result is the following. Suppose \(G\subset \operatorname{Aut}(V)\) is an abelian subgroup. Then the set \(U\) of elements in \(G\) of zero entropy is a subgroup and \(G \simeq U\times H\), where \(H\) is a subgroup of \(G\) with all elements of \(H - \{ e \}\) being of positive entropy.
The authors also show that \(H\) is a free abelian group with \(\text{rank}(H) \leq n-1\) (the estimate is sharp) and if \(\text{rank}(H) = n-1\), then \(U\) is finite.
Other inequalities connecting \(\text{rank}(H)\) and the dimensions of various Dolbeault cohomology groups of \(V\) are also derived.

MSC:
32M05 Complex Lie groups, group actions on complex spaces
32Q15 Kähler manifolds
37B40 Topological entropy
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[1] S. Bochner \et D. Montgomery, Groups of differentiable and real or complex analytic transformations , Ann. of Math. (2) 46 (1945), 685-694. JSTOR: · Zbl 0061.04406
[2] D. Burns Jr., S. Shnider, \et R. O. Wells Jr., Deformations of strictly pseudoconvex domains , Invent. Math. 46 (1978), 237-253. · Zbl 0412.32022
[3] S. Cantat, Dynamique des automorphismes des surfaces K3 , Acta Math. 187 (2001), 1-57. · Zbl 1045.37007
[4] J.-P. Demailly,“Monge-ampère operators, Lelong numbers and intersection theory” dans Complex Analysis and Geometry , Univ. Ser. Math., Plenum, New York, 1993, 115-193. · Zbl 0792.32006
[5] J.-P. Demailly \et M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold , prépublication. · Zbl 1064.32019
[6] T.-C. Dinh \et N. Sibony, Sur les endomorphismes holomorphes permutables de \(\mathbbP^k\) , Math. Ann. 324 (2002), 33-70. · Zbl 1090.32009
[7] —-, Une borne supérieure pour l’entropie topologique d’une application rationnelle , prépublication.
[8] A. E. Eremenko, Some functional equations connected with iteration of rational functions , Leningrad. Math. J. 1 , no. 4 (1990), 905-919. · Zbl 0724.39006
[9] P. Eyssidieux, Un théorème de Nakai-Moishezon pour certaines classes de type \((1,1)\) , prépublication. · Zbl 1332.14016
[10] P. Fatou, Sur l’itération analytique et sur les substitutions permutables , J. Math. Pures Appl. (9) 2 (1923), 343-384. · JFM 49.0712.03
[11] S. Friedland, “Entropy of algebraic maps” dans Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993) , J. Fourier Anal. Appl. 1995 , Special Issue, 215-228. · Zbl 0890.54018
[12] P. Griffiths \et J. Harris, Principles of Algebraic Geometry , reimpression de l’original (1978), Wiley Classics Lib., Wiley, New York, 1994. · Zbl 0408.14001
[13] M. Gromov, Entropy, homology and semialgebraic geometry , Astérique 145 -. 146 (1987), 5, 225-240., Séminaire Bourbaki 1985/86, exp. 663. · Zbl 0611.58041
[14] -. -. -. -., “Convex sets and Kähler manifolds” dans Advances in Differential Geometry and Topology , World Sci., Teaneck, N.J., 1990, 1-38.
[15] -. -. -. -., On the entropy of holomorphic maps , manuscrit (1977), Enseign. Math. (2) 49 (2003), 217-235.
[16] J. E. Humphreys, Linear Algebraic Groups , Grad. Texts in Math. 21 , Springer, New York, 1975.
[17] G. Julia, Mémoire sur la permutabilité des fractions rationnelles , Ann. Sci. École Norm. Sup. (3) 39 (1922), 131-215. · JFM 48.0364.02
[18] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems , Encyclopedia Math. Appl. 54 , Cambridge Univ. Press, Cambridge, 1995. · Zbl 0878.58020
[19] A. Lamari, Le cône kählérien d’une surface , J. Math. Pures Appl. (9) 78 (1999), 249-263. · Zbl 0941.32007
[20] S. Lamy, L’alternative de Tits pour \(\mathrm Aut(\mathbbC^2)\) , J. Algebra 239 (2001), 413-437. · Zbl 1040.37031
[21] D. I. Lieberman, “Compactness of the Chow scheme: Applications to automorphisms and deformations of Kähler manifolds” dans Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975-1977.) , Lecture Notes in Math. 670 , Springer, Berlin, 1978, 140-186. · Zbl 0391.32018
[22] B. Mazur, The topology of rational points , Experiment. Math. 1 (1992), 35-45. · Zbl 0784.14012
[23] C. T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks , J. Reine Angew. Math. 545 (2002), 201-233. · Zbl 1054.37026
[24] G. Prasad \et M. S. Raghunathan, Cartan subgroups and lattices in semi-simple groups , Ann. of Math. (2) 96 (1972), 296-317. JSTOR: · Zbl 0245.22013
[25] J. F. Ritt, Permutable rational functions , Trans. Amer. Math. Soc. 25 (1923), 399-448. JSTOR: · JFM 49.0712.02
[26] P. Samuel, Théorie algébrique des nombres , Hermann, Paris, 1967. · Zbl 0146.06402
[27] S. Smale, Dynamics retrospective: Great problems, attempts that failed , Phys. D 51 (1991), 267-273. · Zbl 0745.58018
[28] A. P. Veselov, Integrable mappings and Lie algebras (en russe), Dokl. Akad. Nauk SSSR 292 , no. 6 (1987), 1289-1291.; Traduction anglaise dans Soviet Math. Dokl. 35 , no. 1 (1987), 211-213.
[29] V. A. Timorin, Mixed Hodge-Riemann bilinear relations in a linear context (en russe), Funktsional. Anal. i Prilozhen. 32 , no. 4 (1998), 63-68.; Traduction anglaise dans Funct. Anal. Appl. 32 , no. 4 (1998), 268-272. · Zbl 0948.32021
[30] Y. Yomdin, Volume growth and entropy , Israel J. Math. 57 (1987), 285-300. · Zbl 0641.54036
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