Acta Mathematica Sinica, English Series

, Volume 34, Issue 8, pp 1208–1224 | Cite as

A Cartan’s Second Main Theorem Approach in Nevanlinna Theory



In 2002, in the paper entitled “A subspace theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.


Nevanlinna theory the Second Main Theorem 

MR(2010) Subject Classification

32H30 32H22 11J97 


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  1. [1]
    Autissier, P.: Sur la non-densité des points entiers (in French) [On the nondensity of integral points]. Duke Math. J., 158(1), 13–27 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Cartan, H.: Sur les zeros des combinaisions linearires de p fonctions holomorpes donnees. Mathematica (Cluj), 7, 80–103 (1933)Google Scholar
  3. [3]
    Corvaja, P., Zannier, U.: A subspace theorem approach to integral points on curves. C. R. Acad. Sci. Paris, 334, 267–271 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Corvaja, P., Zannier, U.: On integral points on surfaces. Ann. of Math., 160(2), 705–726 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Corvaja, P., Zannier, U.: On a general Thue’s equation. Amer. J. Math., 126(5), 1033–1055 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Evertse, J. H., Ferretti, R. G.: Diophantine inequalities on projective varieties. Int. Math. Res. Notices, 25, 1295–1330 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Evertse, J. H., Ferretti, R.: A generalization of the Subspace Theorem with polynomials of higher degree. In Diophantine approximation, volume 16 of Dev. Math., pages 175–198, Springer Wien NewYork, Vienna, 2008Google Scholar
  8. [8]
    Faltings, G., Wüstholz, G.: Diophantine approximations on projective varieties. Invent. Math., 116, 109–138 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Hussein, S., Ru, M.: A general defect relation and height inequality for divisors in subgeneral position, to appear in Asian J. of Math.Google Scholar
  10. [10]
    Levin, A.: Generalizations of Siegel’s and Picard’s theorems. Ann. of Math. (2), 170(2), 609–655 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    McKinnon, D., Roth, M.: Seshadri constants, diophantine approximation and Roth’s theorem for arbitrary varieties. Invent. Math., 200, 513–583 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Nevanlinna, R.: Zur Theorie der meromorphen Funktionen. Acta Math., 46, 1–99 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Ru, M.: Nevanlinna Theory and Its Relation to Diophantine Approximation, World Scientific, Singapore, 2001CrossRefzbMATHGoogle Scholar
  14. [14]
    Ru, M.: On the general form of the second main theorem. Trans. Amer. Math. Soc., 349, 5093–5105 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Ru, M.: A defect relation for holomorphic curves intersecting hypersurfaces. Amer. J. Math., 126, 215–226 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Ru, M.: Holomorphic curves into algebraic varieties. Ann. of Math., 169, 255–267 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Ru, M.: A defect relation for holomorphic curves intersecting general divisors in projective varieties. J. Geometric Anal., 26(4), 2751–2776 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Ru, M.: A general Diophantine inequality. Funct. Approx., 56(2), 143–163 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Ru, M.: Recent progress in Nevanlinna theory. J. Jiangxi Normal Univ. (Natural Sci. Ed.), 42(1), 1–11 (2018)Google Scholar
  20. [20]
    Ru, M., Vojta, P.: Birational Nevanlinna constant and its consequences, Preprint, arXiv:1608.05382 [math.NT]Google Scholar
  21. [21]
    Ru, M., Wang, J.: A subspace theorem for subvarieties. Algebra Number Theory, 11(10), 2323–2337 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Vojta, P.: Diophantine Approximation and Value Distribution Theory, Volume 1239 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1987CrossRefzbMATHGoogle Scholar
  23. [23]
    Weyl, H.: Meromorphic Functions and Analytic Curves, Princeton Univ. Press, Princeton, NJ, 1943zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsEast China Normal UniversityShanghaiP. R. China
  2. 2.Department of MathematicsUniversity of HoustonHoustonUSA

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