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CARLSON–GRIFFITHS THEORY FOR COMPLETE KÄHLER MANIFOLDS

Part of: Holomorphic mappings and correspondences Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  14 February 2022

Xianjing Dong Open the ORCID record for Xianjing Dong [Opens in a new window]
Xianjing Dong*
Affiliation:
School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P. R. China
*
xjdong@amss.ac.cn
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Abstract

We investigate Carlson–Griffiths’ equidistribution theory of meormorphic mappings from a complete Kähler manifold into a complex projective algebraic manifold. By using a technique of Brownian motions developed by Atsuji, we obtain a second main theorem in Nevanlinna theory provided that the source manifold is of nonpositive sectional curvature. In particular, a defect relation follows if some growth condition is imposed.

Keywords

Nevanlinna theoryvalue distributionsecond main theoremlogarithmic derivative lemmadefect relation

MSC classification

Primary: 30D35: Distribution of values, Nevanlinna theory
Secondary: 32H30: Value distribution theory in higher dimensions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 5 , September 2023 , pp. 2337 - 2365
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

Atsuji, A., A second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds, J. Math. Soc. Japan 60 (2008), 471–493.CrossRefGoogle Scholar
Atsuji, A., Estimates on the number of the omitted values by meromorphic functions, Adv. Stud. Pure Math. 57 (2010), 49–59.10.2969/aspm/05710049CrossRefGoogle Scholar
Atsuji, A., On the number of omitted values by a meromorphic function of finite energy and heat diffusions, J. Geom. Anal. 20 (2010), 1008–1025.CrossRefGoogle Scholar
Atsuji, A., Nevanlinna-type theorems for meromorphic functions on non-positively curved Kähler manifolds, Forum Math. 30 (2018), 171–189.CrossRefGoogle Scholar
Bass, R. F., Probabilistic Techniques in Analysis (Springer, New York, 1995).Google Scholar
Bishop, R. L. and Crittenden, R. J., Geometry of Manifolds (Academic Press, New York, 1964).Google Scholar
Carlson, J. and Griffiths, P., A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557–584.CrossRefGoogle Scholar
Carne, T. K., Brownian motion and Nevanlinna theory, Proc. London Math. Soc. (3) 52 (1986), 349–368.CrossRefGoogle Scholar
Chung, K. L., Lectures from Markov Processes to Brownian Motion , Grund. Math. Wiss., Vol. 24 (Springer-Verlag, New York, Heidelberg, Berlin, 1982).Google Scholar
Debiard, A., Gaveau, B. and Mazet, E., Theorems de comparaison en geometrie Riemannienne, Vol. 12 (Publ. Res. Inst. Math. Sci. Kyoto, (1976), 390–425.Google Scholar
Dong, X. J., He, Y. and Ru, M., Nevalinna theory through the Brownian motion, Sci. China Math. 62 (2019), 2131–2154.CrossRefGoogle Scholar
Elworthy, K. D., Li, X. M. and Yor, M., On the tails of the supremum and the quadratic variation of strictly local martingales, in LNM 1655, Séminaire de Probabilités, Vol. XXXI (Springer, Berlin, 1997), 113–125.Google Scholar
Griffiths, P., Entire Holomorphic Mappings in One and Several Complex Variables (Princeton University Press, Princeton, NJ, 1976).10.1515/9781400881482CrossRefGoogle Scholar
Griffiths, P. and King, J., Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 146–220.CrossRefGoogle Scholar
Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Ann. Math. 79 (1964), 109–203, 205–326.10.2307/1970486CrossRefGoogle Scholar
Hsu, E. P., Stochastic Analysis on Manifolds , Grad. Stud. Math., Vol. 38 (American Mathematical Society, Providence, RI, 2002).Google Scholar
Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Vol. 24, 2nd edn. (North-Holland Mathematical Library, North-Holland, Amsterdam, (1989).Google Scholar
Itô, K. and McKean, H. P. Jr., Diffusion Processes and Their sample Paths (Academic Press, New York, 1965).Google Scholar
Noguchi, J., Meromorphic mappings of a covering space over ${\mathbb{C}}^m$ into a projective varieties and defect relations, Hiroshima Math. J. 6 (1976), 265–280.Google Scholar
Noguchi, J. and Ochiai, T., Geometric function theory in several variables, Vol. 80 (American Mathematical Society, Providence, RI, 1990).CrossRefGoogle Scholar
Noguchi, J. and Winkelmann, J., Nevanlinna theory in several complex variables and Diophantine approximation, in A Series of Comprehensive Studies in Mathematics (Springer, 2014).Google Scholar
Port, S. C. and Stone, C. J., Brownian Motion and Classical Potential Theory (Academic Press, New York, 1978).Google Scholar
Ru, M., Nevanlinna Theory and Its Relation to Diophantine Approximation, 1st edn. (World Scientific Publishing Company, 2001).CrossRefGoogle Scholar
Sakai, F., Degeneracy of holomorphic maps with ramification, Invent. Math. 26 (1974), 213–229.10.1007/BF01418950CrossRefGoogle Scholar
Shiffman, B., Nevanlinna defect relations for singular divisors, Invent. Math. 31 (1975), 155–182.CrossRefGoogle Scholar