Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 455–470 | Cite as

Second Main Theorem and Unicity of Meromorphic Mappings for Hypersurfaces in Projective Varieties

  • Si Duc Quang
  • Do Phuong An


Let V be a projective subvariety of \(\mathbb P^{n}(\mathbb C)\). A family of hypersurfaces \(\{Q_{i}\}_{i=1}^{q}\) in \(\mathbb P^{n}(\mathbb C)\) is said to be in N-subgeneral position with respect to V if for any 1≤i 1<⋯<i N+1q, \( V\cap (\bigcap _{j=1}^{N+1}Q_{i_{j}})=\varnothing \). In this paper, we will prove a second main theorem for meromorphic mappings of \(\mathbb C^{m}\) into V intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give a uniqueness theorem for meromorphic mappings of \(\mathbb C^{m}\) into V sharing a few hypersurfaces without counting multiplicity. In particular, we extend the uniqueness theorem for linearly nondegenerate meromorphic mappings of \(\mathbb C^{m}\) into \(\mathbb P^{n}(\mathbb C)\) sharing 2n+3 hyperplanes in general position to the case where the mappings may be linearly degenerated.


Holomorphic curves Algebraic degeneracy Defect relation Nochka weight 

Mathematics Subject Classification (2010)

Primary 32H30 Secondary 32H04 32H25 14J70 



This work was completed while the first author was staying at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the Institute for the support.

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2015.03.


  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. Houston J. Math. 35, 775–786 (2009)MathSciNetMATHGoogle Scholar
  2. 2.
    An, D.P., Quang, S.D., Thai, D.D.: The second main theorem for meromorphic mappings into a complex projective space. Acta. Math. Vietnam. 38(1), 187–205 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, Z., Yan, Q.: Uniqueness theorem of meromorphic mappings into \(\mathbb P^{n}(\mathbb C)\) sharing 2N+3 hyperplanes regardless of multiplicities. Internat. J. Math. 20, 717–726 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dulock, M., Ru, M.: A uniqueness theorem for holomorphic curves sharing hypersurfaces. Complex Var. Elliptic Equ. 53, 797–802 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dethloff, G., Tan, T.V.: A uniqueness theorem for meromorphic maps with moving hypersurfaces. Publ. Math. Debrecen. 78, 347–357 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Fujimoto, H.: Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into \(\mathbb P^{N_{1}}(\mathbb C)\times {\dots } \times \mathbb P^{N_{k}}(\mathbb C)\). Jpn. J. Math. 11, 233–264 (1985)Google Scholar
  7. 7.
    Nochka, E.I.: On the theory of meromorphic functions. Sov. Math. Dokl. 27, 377–381 (1983)MATHGoogle Scholar
  8. 8.
    Noguchi, J.: A note on entire pseudo-holomorphic curves and the proof of Cartan-Nochka’s theorem. Kodai Math. J. 28, 336–346 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ru, M.: Holomorphic curves into algebraic varieties. Ann. Math. 169, 255–267 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Shiffman, B.: Introduction to the Carlson-Griffiths equidistribution theory. Lect. Notes Math. 981, 44–89 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Division of MathematicsBanking AcademyHanoiVietnam

Personalised recommendations