Differentiably Non-degenerate Meromorphic Maps

  • Junjiro Noguchi
  • Jörg Winkelmann
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 350)


At the beginning of the 1970s, P.A. Griffiths et al. extended Nevanlinna theory to the higher dimensional case, and established the theory for differentiably non-degenerate holomorphic mappings f:WV from an affine algebraic variety W into a projective algebraic variety V with \(\operatorname {rank}df = \dim V\). This theory was very different to the Nevanlinna–Weyl–Ahlfors theory extended by W. Stoll, and was very fresh. The theory has been generalized in various ways, including the case of meromorphic mappings, applications have been developed, and a new phase was brought into the value distribution theory.


Line Bundle Meromorphic Mapping Compact Complex Manifold Kodaira Dimension Normal Complex Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Barth, W., Peters, C. and Van de Ven, A.

  1. [84]
    Compact Complex Surfaces, Ergebnisse Math. und ihrer Grenzgebiets 4, Springer, Berlin, 1984. CrossRefMATHGoogle Scholar

Biancofiore, A. and Stoll, W.

  1. [81]
    Another proof of the lemma of the logarithmic derivative in several complex variables, Recent Developments in Several Complex Variables, Ann. Math. Studies 100, pp. 29–45, Princeton University Press, Princeton, 1981. Google Scholar

Bieberbach, L.

  1. [33]
    Beispiel zweier ganzer Funktionen zweier komplexer Variablen, welche eine schlicht volumentreue Abbildung des R 4 auf einen Teil seiner selbst vermitteln, Sitzungsber. Preuss. Akad. Wiss., Phy.-Math. 14–15 (1933), 476–479. Google Scholar

Buzzard, G.T. and Lu, S.S.Y.

  1. [00]
    Algebraic surfaces holomorphically dominable by C 2, Invent. Math. 139 (2000), 617–659. MathSciNetCrossRefMATHGoogle Scholar

Carlson, J. and Griffiths, P.

  1. [72]
    A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. Math. 95 (1972), 557–584. MathSciNetCrossRefMATHGoogle Scholar

Cornalba, M. and Shiffman, B.

  1. [72]
    A counterexample to the “Transcendental Bezout problem”, Ann. Math. 96 (1972), 402–406. MathSciNetCrossRefMATHGoogle Scholar

Fatou, P.

  1. [22]
    Sur les fonctions méromorphes de deux variables, C. R. Math. Acad. Sci. Paris 175 (1922), 862–865. MATHGoogle Scholar

Green, M.L.

  1. [78]
    Holomorphic maps to complex tori, Am. J. Math. 100 (1978), 615–620. CrossRefMATHGoogle Scholar

Griffiths, P. and King, J.

  1. [73]
    Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220. MathSciNetCrossRefMATHGoogle Scholar

Kodaira, K.

  1. [68]
    On the structure of compact complex analytic surfaces, IV, Am. J. Math. 90 (1968), 1048–1066. MathSciNetCrossRefMATHGoogle Scholar
  2. [71]
    Holomorphic mappings of polydiscs into compact complex manifolds, J. Differ. Geom. 6 (1971), 33–46. MathSciNetMATHGoogle Scholar
  3. [75]
    Collected Works/Kunihiko Kodaira, I–III, Iwanami Shoten/Princeton University Press, Tokyo/Princeton, 1975. Google Scholar

Noguchi, J.

  1. [76a]
    Meromorphic mappings of a covering space over C m into a projective variety and defect relations, Hiroshima Math. J. 6 (1976a), 265–280. MathSciNetMATHGoogle Scholar

Noguchi, J. and Ochiai, T.

  1. [90]
    Geometric Function Theory in Several Complex Variables, Transl. Math. Mono. 80, Am. Math. Soc., Providence, 1990. MATHGoogle Scholar

Noguchi, J. and Winkelmann, J.

  1. [12]
    Order of meromorphic maps and rationality of the image space, J. Math. Soc. Jpn. 64 no. 4 (2012), 1169–1180. MathSciNetCrossRefMATHGoogle Scholar

Ochiai, T. and Noguchi, J.

  1. [84]
    Geometric Function Theory in Several Variables (in Japanese), Iwanami Shoten, Tokyo, 1984. Google Scholar

Pjateckiĭ-Šapiro, I.I. and Shafarevič, I.R.

  1. [71]
    Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR, Ser. Mat. 35 (1971), 530–572. MathSciNetGoogle Scholar

Ru, M.

  1. [01]
    Nevanlinna Theory and Its Relation to Diophantine Approximation, World Scientific, Singapore, 2001. CrossRefMATHGoogle Scholar

Sakai, F.

  1. [74a]
    Degeneracy of holomorphic maps with ramification, Invent. Math. 26 (1974a), 213–229. MathSciNetCrossRefMATHGoogle Scholar
  2. [74b]
    Defect relations and ramifications, Proc. Jpn. Acad. 50 (1974b), 723–728. CrossRefMATHGoogle Scholar
  3. [76]
    Defect relations for equidimensional holomorphic maps, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 23 (1976), 561–580. MATHGoogle Scholar

Selberg, H.L.

  1. [30]
    Über die Wertverteilung der algebroïden Funktionen, Math. Z. 31 (1930), 709–728. MathSciNetCrossRefMATHGoogle Scholar
  2. [34]
    Algebroïde Funktionen und Umkehrfunktionen Abelscher Integrale, Avh. - Nor. Vidensk.-Akad. Oslo 8 (1934), 1–72. Google Scholar

Shiffman, B.

  1. [75]
    Nevanlinna defect relations for singular divisors, Invent. Math. 31 (1975), 155–182. MathSciNetCrossRefMATHGoogle Scholar

Stoll, W.

  1. [77a]
    Value Distribution on Parabolic Spaces, Lect. Notes Math. 600, Springer, Berlin, 1977a. MATHGoogle Scholar

Valiron, G.

  1. [29]
    Sur les fonctions algébroïdes méromorphes du second degré, C. R. Math. Acad. Sci. Paris 189 (1929), 623–625. MATHGoogle Scholar
  2. [31]
    Sur la dérivée des fonctions algébroïdes, Bull. Soc. Math. Fr. 59 (1931), 17–39. MathSciNetGoogle Scholar

Vitter, A.L.

  1. [77]
    The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), 89–104. MathSciNetCrossRefMATHGoogle Scholar

Weil, A.

  1. [58]
    Introduction à l’Étude des Variétés Kähleriennes, Hermann, Paris, 1958. MATHGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Junjiro Noguchi
    • 1
    • 2
  • Jörg Winkelmann
    • 3
  1. 1.The University of TokyoTokyoJapan
  2. 2.Tokyo Institute of TechnologyTokyoJapan
  3. 3.Ruhr-University BochumBochumGermany

Personalised recommendations