Chinese Annals of Mathematics, Series B

, Volume 40, Issue 2, pp 251–272 | Cite as

Degeneracy and Finiteness Theorems for Meromorphic Mappings in Several Complex Variables

  • Si Duc QuangEmail author


The author proves that there are at most two meromorphic mappings of ℂm into ℙn(ℂ) (n ≥ 2) sharing 2n+2 hyperplanes in general position regardless of multiplicity, where all zeros with multiplicities more than certain values do not need to be counted. He also shows that if three meromorphic mappings f1, f2, f3 of ℂm into ℙn(ℂ) (n ≥ 5) share 2n+1 hyperplanes in general position with truncated multiplicity, then the map f1×f2×f3 is linearly degenerate.


Second main theorem Uniqueness problem Meromorphic mapping Multiplicity 

2000 MR Subject Classification

32H30 32A22 30D35 


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  1. [1]
    Chen, Z. and Yan, Q., Uniqueness theorem of meromorphic mappings into ℙn(ℂ) sharing 2N+3 hyperplanes regardless of multiplicities, Internat. J. Math., 20, 2009, 717–726.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Dethloff, G. and Tan, T. V., Uniqueness theorems for meromorphic mappings with few hyperplanes, Bull. Sci. Math., 133, 2009, 501–514.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Fujimoto, H., Uniqueness problem with truncated multiplicities in value distribution theory, Nagoya Math. J., 152, 1998, 131–152.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Nevanlinna, R., Einige eideutigkeitss¨atze in der theorie der meromorphen funktionen, Acta. Math., 48, 1926, 367–391.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Noguchi, J. and Ochiai, T., Introduction to Geometric Function Theory in Several Complex Variables, Trans. Math. Monogr., 80, Amer. Math. Soc., Providence, RI, 1990.CrossRefzbMATHGoogle Scholar
  6. [6]
    Quang, S. D., Unicity of meromorphic mappings sharing few hyperplanes, Ann. Polon. Math., 102(3), 2011, 255–270.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Quang, S. D., A finiteness theorem for meromorphic mappings sharing few hyperplanes, Kodai Math. J., 102(35), 2012, 463–484.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Quang, S. D. and Quynh, L. N., Algebraic dependences of meromorphic mappings sharing few hyperplanes counting truncated multiplicities, Kodai Math. J., 38, 2015, 97–118MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Smiley, L., Geometric conditions for unicity of holomorphic curves, Contemp. Math., 25, 1983, 149–154.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Thai, D. D. and Quang, S. D., Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables, Int. J. Math., 17(10), 2006, 1223–1257.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Yan, Q., and Chen, Z., Degeneracy theorem for meromorphic mappings with truncated multiplicity, Acta Math. Scientia, 31B, 2011, 549–560.zbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationCau Giay, HanoiVietnam
  2. 2.Thang Long Institute of Mathematics and Applied SciencesNghiem Xuan Yem, Hoang Mai, Ha NoiVietnam

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