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Green currents for holomorphic automorphisms of compact Kähler manifolds. (English) Zbl 1066.32024
Authors’ abstract: Let \(f\) be a holomorphic automorphism of a compact Kähler manifold \((X,\omega)\) of dimension \(k\geq 2\). We study the convex cones of positive closed \((p,p)\)-currents \(T_p\), which satisfy a functional relation \[ f^* T_p=\lambda T_p, \lambda>1, \] and some regularity condition (PB, PC). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy \({h}(f)\) of \(f\) is positive, then for some \(m\geq 1\), there is a positive closed \((m,m)\)-current \(T_m\) which satisfies the relation \[ f^* T_m=\exp({h}(f)) T_m. \] Moreover, every quasi-p.s.h. function is integrable with respect to the trace measure of \(T_m\). When the dynamical degrees of \(f\) are all distinct, we construct an invariant measure \(\mu\) as an intersection of closed currents. We show that this measure is mixing and gives no mass to pluripolar sets and to sets of small Hausdorff dimension.
Reviewer: Zhuan Ye (DeKalb)

MSC:
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32Q15 Kähler manifolds
32U40 Currents
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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References:
[1] Eric Bedford, Mikhail Lyubich, and John Smillie, Polynomial diffeomorphisms of \?². IV. The measure of maximal entropy and laminar currents, Invent. Math. 112 (1993), no. 1, 77 – 125. · Zbl 0792.58034
[2] Eric Bedford and John Smillie, Polynomial diffeomorphisms of \?². III. Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann. 294 (1992), no. 3, 395 – 420. · Zbl 0765.58013
[3] André Blanchard, Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. (3) 73 (1956), 157 – 202 (French). · Zbl 0073.37503
[4] J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), no. 4, 903 – 1027. · Zbl 0973.14013
[5] Jean-Yves Briend and Julien Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de \?^{\?}(\?), Publ. Math. Inst. Hautes Études Sci. 93 (2001), 145 – 159 (French, with English and French summaries). · Zbl 1010.37004
[6] Serge Cantat, Dynamique des automorphismes des surfaces \?3, Acta Math. 187 (2001), no. 1, 1 – 57 (French). · Zbl 1045.37007
[7] Laurent Clozel and Emmanuel Ullmo, Correspondances modulaires et mesures invariantes, J. Reine Angew. Math. 558 (2003), 47 – 83 (French). · Zbl 1042.11027
[8] Jean-Pierre Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, pp. 115 – 193. · Zbl 0792.32006
[9] Jean-Pierre Demailly, Théorie de Hodge \?² et théorèmes d’annulation, Introduction à la théorie de Hodge, Panor. Synthèses, vol. 3, Soc. Math. France, Paris, 1996, pp. 3 – 111 (French).
[10] Jean-Pierre Demailly, Pseudoconvex-concave duality and regularization of currents, Several complex variables (Berkeley, CA, 1995 – 1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 233 – 271. · Zbl 0960.32011
[11] T.C. Dinh, Distribution des préimages et des points périodiques d’une correspondance polynomiale, Bull. Soc. Math. France, to appear.
[12] T.C. Dinh, Suites d’applications méromorphes multivaluées et courants laminaires, preprint, 2003. arXiv:math.DS/0309421.
[13] Tien-Cuong Dinh and Nessim Sibony, Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9) 82 (2003), no. 4, 367 – 423 (French, with English and French summaries). · Zbl 1033.37023
[14] Tien-Cuong Dinh and Nessim Sibony, Dynamique des applications polynomiales semi-régulières, Ark. Mat. 42 (2004), no. 1, 61 – 85 (French, with English summary). · Zbl 1059.37033
[15] Tien-Cuong Dinh and Nessim Sibony, Groupes commutatifs d’automorphismes d’une variété kählérienne compacte, Duke Math. J. 123 (2004), no. 2, 311 – 328 (French, with English and French summaries). · Zbl 1065.32012
[16] T.C. Dinh and N. Sibony, Distribution de valeurs d’une suite de transformations méromorphes et applications, preprint, 2003. arXiv:math.DS/0306095.
[17] T.C. Dinh and N. Sibony, Une borne supérieure de l’entropie topologique d’une application rationnelle, Ann. of Math., to appear.
[18] T.C. Dinh and N. Sibony, Regularization of currents and entropy, Ann. Sci. Ecole Norm. Sup., to appear. · Zbl 1074.53058
[19] T.C. Dinh and N. Sibony, Dynamics of regular birational maps in \(\mathbb{P} ^k\), J. Funct. Anal., to appear. · Zbl 1067.37055
[20] T.C. Dinh and N. Sibony, Decay of correlations and central limit theorem for meromorphic maps, preprint, 2004. arXiv:math.DS/0410008.
[21] Charles Favre and Vincent Guedj, Dynamique des applications rationnelles des espaces multiprojectifs, Indiana Univ. Math. J. 50 (2001), no. 2, 881 – 934 (French, with English summary). · Zbl 1046.37026
[22] John Erik Fornæss and Nessim Sibony, Complex Hénon mappings in \?² and Fatou-Bieberbach domains, Duke Math. J. 65 (1992), no. 2, 345 – 380. · Zbl 0761.32015
[23] John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimensions, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 131 – 186. Notes partially written by Estela A. Gavosto. · Zbl 0811.32019
[24] Henri Gillet and Christophe Soulé, Arithmetic intersection theory, Inst. Hautes Études Sci. Publ. Math. 72 (1990), 93 – 174 (1991). · Zbl 0741.14012
[25] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. · Zbl 0836.14001
[26] Mikhaïl Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), no. 3-4, 217 – 235. · Zbl 1080.37051
[27] M. Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1 – 38. · Zbl 0770.53042
[28] Vincent Guedj, Dynamics of polynomial mappings of \Bbb C², Amer. J. Math. 124 (2002), no. 1, 75 – 106. · Zbl 1198.32007
[29] V. Guedj, Ergodic properties of rational mappings with large topological degree, Ann. of Math., to appear. · Zbl 1088.37020
[30] Vincent Guedj and Nessim Sibony, Dynamics of polynomial automorphisms of \?^{\?}, Ark. Mat. 40 (2002), no. 2, 207 – 243. · Zbl 1034.37025
[31] A. G. Khovanskiĭ, Fewnomials and Pfaff manifolds, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 549 – 564. · Zbl 0586.51015
[32] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. · Zbl 0732.22008
[33] Barry Mazur, The topology of rational points, Experiment. Math. 1 (1992), no. 1, 35 – 45. · Zbl 0784.14012
[34] Curtis T. McMullen, Dynamics on \?3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math. 545 (2002), 201 – 233. · Zbl 1054.37026
[35] Nessim Sibony, Dynamique des applications rationnelles de \?^{\?}, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix – x, xi – xii, 97 – 185 (French, with English and French summaries). · Zbl 1020.37026
[36] B. Teissier, Bonnesen-type inequalities in algebraic geometry. I. Introduction to the problem, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 85 – 105. · Zbl 0494.52009
[37] C. Voisin, Intrinsic pseudovolume forms and K-correspondences, preprint, 2003, arXiv: math.AG/0212110.
[38] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285 – 300. · Zbl 0641.54036
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