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On the Weyl–Ahlfors theory of derived curves

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Abstract

For derived curves intersecting a family of decomposable hyperplanes in subgeneral position, we obtain an analog of the Cartan–Nochka Second Main Theorem, generalizing a classical result of Fujimoto about decomposable hyperplanes in general position.

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References

  1. Ahlfors, L.V.: The theory of meromorphic curves. Acta Soc. Sci. Fennicae. Nova Ser. A 3(4), 31 (1941)

    MathSciNet  MATH  Google Scholar 

  2. Cartan, H.: Sur les zéros des combinaisons linéarires de \(p\) fonctions holomorphes données. Mathematica 7, 5–31 (1933)

    MATH  Google Scholar 

  3. Chen, W.X.: Defect relations for degenerate meromorphic maps. Trans. Am. Math. Soc. 319(2), 499–515 (1990)

    Article  MathSciNet  Google Scholar 

  4. Cowen, M., Griffiths, P.: Holomorphic curves and metrics of negative curvature. J. Anal. Math. 29, 93–153 (1976)

    Article  MathSciNet  Google Scholar 

  5. Fujimoto, H.: The defect relations for the derived curves of a holomorphic curve in \(P^{n}({ C})\). Tohoku Math. J. (2) 34(1), 141–160 (1982)

    Article  MathSciNet  Google Scholar 

  6. Fujimoto, H.: Value distribution theory of the Gauss map of minimal surfaces in \({ R}^m\). Aspects of Mathematics, E21. Friedr. Vieweg & Sohn, Braunschweig (1993)

    Book  Google Scholar 

  7. Nevanlinna, R.: Zur Theorie der Meromorphen Funktionen. Acta Math. 46(1–2), 1–99 (1925)

    Article  MathSciNet  Google Scholar 

  8. Nochka, E.I.: On the theory of meromorphic curves. Dokl. Akad. Nauk SSSR 269(3), 547–552 (1983)

    MathSciNet  MATH  Google Scholar 

  9. Noguchi, J., Winkelmann, J.: Nevanlinna theory in several complex variables and Diophantine approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 350. Springer, Tokyo (2014)

    Book  Google Scholar 

  10. Ru, M.: Holomorphic curves into algebraic varieties. Ann. Math. (2) 169(1), 255–267 (2009)

    Article  MathSciNet  Google Scholar 

  11. Ru, M.: A Cartan’s second main theorem approach in Nevanlinna theory. Acta Math. Sin. (Engl. Ser.) 34(8), 1208–1224 (2018)

    Article  MathSciNet  Google Scholar 

  12. Shabat, B.V.: Distribution of values of holomorphic mappings, volume 61 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1985. Translated from the Russian by J. R. King, Translation edited by Lev J. Leifman

  13. Stoll, W.: Die beiden Hauptsätze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veränderlichen. I. Acta Math. 90, 1–115 (1953)

    Article  MathSciNet  Google Scholar 

  14. Stoll, W.: Die beiden Hauptsätze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veränderlichen. II. Acta Math. 92, 55–169 (1954)

    Article  MathSciNet  Google Scholar 

  15. Weyl, H.: Meromorphic functions and analytic curves. Annals of Mathematics Studies, no. 12. Princeton University Press, Princeton (1943)

    MATH  Google Scholar 

  16. Weyl, H., Weyl, J.: Meromorphic curves. Ann. Math. 39(3), 516–538 (1938)

    Article  MathSciNet  Google Scholar 

  17. Wu, H.-H.: The equidistribution theory of holomorphic curves. Annals of Mathematics Studies, no. 64. Princeton University Press, Princeton; University of Tokyo Press, Tokyo, (1970)

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Acknowledgements

S.-Y. Xie is partially supported by the NSFC Grant No. 11688101. D. T. Huynh is grateful to Academy of Mathematics and Systems Science in Beijing for enhanced scientific ambience and financial support. He also wants to acknowledge partial support from the Core Research Program of Hue University, Grant No. NCM.DHH.2020.15.

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Authors and Affiliations

  1. Hua Loo-Keng center for Mathematical Sciences, Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190, China

    Dinh Tuan Huynh

  2. Department of Mathematics, University of Education, Hue University, 34 Le Loi St., Hue City, Vietnam

    Dinh Tuan Huynh

  3. Hua Loo-Keng Key Laboratory of Mathematics and Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100190, China

    Song-Yan Xie

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  1. Dinh Tuan Huynh
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  2. Song-Yan Xie
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Correspondence to Dinh Tuan Huynh.

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Song-Yan Xie partially supported by NSFC Grant No. 11688101.

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Huynh, D.T., Xie, SY. On the Weyl–Ahlfors theory of derived curves. Math. Z. 300, 475–491 (2022). https://doi.org/10.1007/s00209-021-02798-4

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  • DOI: https://doi.org/10.1007/s00209-021-02798-4

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