Limit currents and value distribution of holomorphic maps

Burns, Daniel; Sibony, Nessim

Annales de l'Institut Fourier, Tome 62 (2012) no. 1, pp. 145-176.

Nous introduisons des courants positifs $d$-fermés ou $d{d}^c$-fermés associés à une application holomorphe $\phi$ entre deux variétés complexes. Les courants sont de bidegré $(p,p)$ selon les indicateurs de croissance de $\phi$. Ce sont les analogues des courants d’Ahfors associés aux applications de $ℂ$ dans une variété $Y$. Nous donnons quelques applications à la théorie de distribution de valeurs.

We construct $d$-closed and $d{d}^c$-closed positive currents associated to a holomorphic map $\phi$ 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.

Classification : 32A22, 32H25, 32H30
Keywords: Ahlfors currents, Brody’s theorem, value distribution theory, equidistribution
Mot clés : Courants d’Ahlfors, théorème de Brody, théorie de la distribution de valeurs, équidistribution

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

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

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[11] Gruman, Lawrence The area of analytic varieties in ${ℂ}^n$, Math. Scand., Volume 41 (1977) no. 2, pp. 365-397 | EuDML | MR | Zbl

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

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

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

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

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

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

[18] Sibony, Nessim Dynamique des applications rationnelles de ${\mathbb{P}}^k$, Dynamique et géométrie complexes (Lyon, 1997) (Panor. Synthèses), Volume 8, Soc. Math. France, Paris, 1999, p. ix-x, xi–xii, 97–185 | MR | Zbl

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

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

[21] de Thélin, Henry Ahlfors’ currents in higher dimension, Ann. Fac. Sci. Toulouse Math. (6), Volume 19 (2010) no. 1, pp. 121-133 | DOI | EuDML | Numdam | MR | Zbl