Limit currents and value distribution of holomorphic maps
Annales de l'Institut Fourier, Volume 62 (2012) no. 1, p. 145-176
We construct d-closed and dd c -closed positive currents associated to a holomorphic map φ via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions. Such classes of currents are also referred to as Ahlfors currents. We give some applications to equidistribution problems in value distribution theory.
Nous introduisons des courants positifs d-fermés ou dd c -fermés associés à une application holomorphe φ entre deux variétés complexes. Les courants sont de bidegré (p,p) selon les indicateurs de croissance de φ. Ce sont les analogues des courants d’Ahfors associés aux applications de dans une variété Y. Nous donnons quelques applications à la théorie de distribution de valeurs.
DOI : https://doi.org/10.5802/aif.2703
Classification:  32A22,  32H25,  32H30
Keywords: Ahlfors currents, Brody’s theorem, value distribution theory, equidistribution
@article{AIF_2012__62_1_145_0,
     author = {Burns, Daniel and Sibony, Nessim},
     title = {Limit currents and value distribution of holomorphic maps},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {1},
     year = {2012},
     pages = {145-176},
     doi = {10.5802/aif.2703},
     zbl = {1252.32002},
     mrnumber = {2986269},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2012__62_1_145_0}
}
Burns, Daniel; Sibony, Nessim. Limit currents and value distribution of holomorphic maps. Annales de l'Institut Fourier, Volume 62 (2012) no. 1, pp. 145-176. doi : 10.5802/aif.2703. https://aif.centre-mersenne.org/item/AIF_2012__62_1_145_0/

[1] Ahlfors, Lars V.; Sario, Leo Riemann surfaces, Princeton University Press, Princeton, N.J., Princeton Mathematical Series, No. 26 (1960) | MR 114911 | Zbl 0196.33801

[2] Bishop, Errett Conditions for the analyticity of certain sets, Michigan Math. J., Tome 11 (1964), pp. 289-304 | Article | MR 168801 | Zbl 0143.30302

[3] Carlson, James A. A moving lemma for the transcendental Bezout problem, Ann. of Math. (2), Tome 103 (1976) no. 2, pp. 305-330 | Article | MR 409901 | Zbl 0321.32008

[4] Chern, Shiing-Shen On holomorphic mappings of hermitian manifolds of the same dimension., Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966), Amer. Math. Soc., Providence, R.I. (1968), pp. 157-170 | MR 234397 | Zbl 0184.31202

[5] Dinh, Tien-Cuong; Sibony, Nessim Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv., Tome 81 (2006) no. 1, pp. 221-258 | Article | MR 2208805

[6] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., Tome 203 (2009) no. 1, pp. 1-82 | Article | MR 2545825

[7] Dinh, Tien-Cuong; Sibony, Nessim Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings, Holomorphic dynamical systems, Springer, Berlin (Lecture Notes in Math.) Tome 1998 (2010), pp. 165-294 | MR 2648690

[8] Fornæss, J. E.; Sibony, N. Harmonic currents of finite energy and laminations, Geom. Funct. Anal., Tome 15 (2005) no. 5, pp. 962-1003 | Article | MR 2221156

[9] Griffiths, Phillip; King, James Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math., Tome 130 (1973), pp. 145-220 | Article | MR 427690 | Zbl 0258.32009

[10] Griffiths, Phillip A. Entire holomorphic mappings in one and several complex variables, Princeton University Press, Princeton, N. J. (1976) (The fifth set of Hermann Weyl Lectures, given at the Institute for Advanced Study, Princeton, N. J., October and November 1974, Annals of Mathematics Studies, No. 85) | MR 447638 | Zbl 0317.32023

[11] Gruman, Lawrence The area of analytic varieties in n , Math. Scand., Tome 41 (1977) no. 2, pp. 365-397 | MR 477126 | Zbl 0376.32008

[12] Gruman, Lawrence La géométrie globale des ensembles analytiques dans n , Séminaire Pierre Lelong-Henri Skoda (Analyse). Années 1978/79 (French), Springer, Berlin (Lecture Notes in Math.) Tome 822 (1980), pp. 90-99 | MR 599020 | Zbl 0446.32007

[13] Hayman, W. K. Meromorphic functions, Clarendon Press, Oxford, Oxford Mathematical Monographs (1964) | MR 164038 | Zbl 0115.06203

[14] Katok, Anatole; Hasselblatt, Boris Introduction to the modern theory of dynamical systems, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 54 (1995) (With a supplementary chapter by Katok and Leonardo Mendoza) | MR 1326374 | Zbl 0878.58020 | Zbl 0878.58019

[15] Mcquillan, Michael Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. (1998) no. 87, pp. 121-174 | Article | MR 1659270 | Zbl 1006.32020

[16] Molzon, Robert E.; Shiffman, Bernard; Sibony, Nessim Average growth estimates for hyperplane sections of entire analytic sets, Math. Ann., Tome 257 (1981) no. 1, pp. 43-59 | Article | MR 630646 | Zbl 0537.32009

[17] Shabat, B. V. Raspredelenie znachenii golomorfnykh otobrazhenii, “Nauka”, Moscow (1982) | MR 701119 | Zbl 0537.32008

[18] Sibony, Nessim Dynamique des applications rationnelles de k , Dynamique et géométrie complexes (Lyon, 1997), Soc. Math. France, Paris (Panor. Synthèses) Tome 8 (1999), p. ix-x, xi–xii, 97–185 | MR 1760844 | Zbl 1020.37026

[19] Stoll, Wilhelm The growth of the area of a transcendental analytic set. I, II, Math. Ann., Tome 156 (1964), pp. 144-170 | Article | MR 166393 | Zbl 0126.09502

[20] Stoll, Wilhelm Value distribution on parabolic spaces, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 600 (1977) | MR 590436 | Zbl 0367.32001

[21] De Thélin, Henry Ahlfors’ currents in higher dimension, Ann. Fac. Sci. Toulouse Math. (6), Tome 19 (2010) no. 1, pp. 121-133 | Article | Numdam | MR 2597784 | Zbl 1195.32004