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Acta Mathematica Vietnamica

, Volume 38, Issue 1, pp 187–205 | Cite as

The second main theorem for meromorphic mappings into a complex projective space

  • Do Phuong An
  • Si Duc Quang
  • Do Duc ThaiEmail author
Article

Abstract

The main purpose of this article is to show the Second Main Theorem for meromorphic mappings of ℂ m into ℙ n (ℂ) intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give two unicity theorems for meromorphic mappings of ℂ m into ℙ n (ℂ) sharing few hypersurfaces without counting multiplicity.

Keywords

Holomorphic curves Algebraic degeneracy Defect relation Nochka weight 

Mathematics Subject Classification (2000)

32H30 32H04 32H25 14J70 

Notes

Acknowledgements

The research of the authors is supported by an NAFOSTED grant of Vietnam (Grant No. 101.01-2011.29).

References

  1. 1.
    An, T.T.H., Phuong, H.T.: An explicit estimate on multiplicity truncation in the second main theorem for holomorphic curves encountering hypersurfaces in general position in projective space. Houst. J. Math. 35(3), 775–786 (2009) zbMATHGoogle Scholar
  2. 2.
    Carlson, J., Griffiths, Ph.: A defect relation for equidimensional holomorphic mappings between algebraic varieties. Ann. Math. 95, 557–584 (1972) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chen, Z., Ru, M., Yan, Q.: The truncated second main theorem and uniqueness theorems. Sci. China Ser. A, Math. 53, 1–10 (2010) MathSciNetGoogle Scholar
  4. 4.
    Dethloff, G.E., Tan, T.V., Thai, D.D.: An extension of the Cartan–Nochka second main theorem for hypersurfaces. Int. J. Math. 22, 863–885 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Do, D.T., Ninh, V.T.: The second main theorem for hypersurfaces. Kyushu J. Math. 65, 219–236 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dulock, M., Ru, M.: A uniqueness theorem for holomorphic curves sharing hypersurfaces. Complex Var. Elliptic Equ. 53, 797–802 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Eremenko, A.E., Sodin, M.L.: The value distribution of meromorphic functions and meromorphic curves from the point of view of potential theory. St. Petersburg Math. J. 3, 109–136 (1992) MathSciNetGoogle Scholar
  8. 8.
    Fujimoto, H.: Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into \(\mathbb{P}^{N_{1}}(\mathbb{C})\times \cdots\times\mathbb{P}^{N_{k}}(\mathbb{C})\). Jpn. J. Math. 11, 233–264 (1985) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Griffiths, P., King, J.: Nevanlinna theory and holomorphic mappings between algebraic varieties. Acta Math. 130, 145–220 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Nochka, E.I.: On the theory of meromorphic functions. Sov. Math. Dokl. 27, 377–381 (1983) zbMATHGoogle Scholar
  11. 11.
    Noguchi, J.: Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties. Nagoya Math. J. 83, 213–233 (1981) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Noguchi, J.: On holomorphic curves in semi-Abelian varieties. Math. Z. 228, 713–721 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Noguchi, J.: A note on entire pseudo-holomorphic curves and the proof of Cartan–Nochka’s theorem. Kodai Math. J. 28, 336–346 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Noguchi, J., Winkelmann, J.: Holomorphic curves and integral points off divisors. Math. Z. 239, 593–610 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Noguchi, J., Winkelmann, J.: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Textbook (2010) Google Scholar
  16. 16.
    Noguchi, J., Winkelmann, J., Yamanoi, K.: The second main theorem for holomorphic curves into semi-Abelian varieties. Acta Math. 188, 129–161 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Noguchi, J., Winkelmann, J., Yamanoi, K.: The second main theorem for holomorphic curves into semiabelian varieties II. Forum Math. 20, 469–503 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ru, M.: A defect relation for holomorphic curves interecting hypersurfaces. Am. J. Math. 126, 215–226 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ru, M.: Holomorphic curves into algebraic varieties. Ann. Math. 169, 255–267 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Shiffman, B.: Introduction to the Carlson-Griffiths Equidistribution Theory. Lecture Notes in Math., vol. 981, pp. 44–89 (1983) Google Scholar
  21. 21.
    Siu, Y.-T.: Defects relations for holomorphic maps between spaces of different dimensions. Duke Math. J. 55, 213–251 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Siu, Y.-T., Yeung, S.-K.: Defects for ample divisors of Abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees. Am. J. Math. 119, 1139–1172 (1997) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam

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