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The First Main Theorem

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Nevanlinna Theory in Several Complex Variables and Diophantine Approximation

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 350))

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Abstract

The value distribution theory with domains in several complex variables was pioneered by Wilhelm Stoll (Math. Z. 57:211–237, 1953a; Acta Math. 90:1–115, 1953b; Acta Math. 92:55–169, 1954). While his presentation may not be familiar or easy to us in modern terminologies, the works which he has contributed, beginning with the integrations over singular analytic subvarieties and the extension of Stokes’ theorem, were fundamental. In the 1960s there were many works on the First Main Theorem; these were summarized by W. Stoll (see in Value Distribution of Holomorphic Maps into Compact Complex Manifolds, 1970, in particular its preface and the listed references). The relation to characteristic classes was made explicit first by Bott–Chern (Acta Math. 114:71–112, 1965). (Readers may find a number of interesting papers on the theory of holomorphic mappings in Chern, Selected Papers, 1978.) In the present chapter we follow Carlson–Griffiths (Ann. Math. 95:557–584, 1972), Griffiths–King (Acta Math. 130:145–220, 1973), Noguchi (Nevanlinna Theory in Several Variables and Diophantine Approximation, 2003b) and Noguchi–Winkelmann–Yamanoi (Forum Math. 20:469–503, 2008) which may be most comprehensive.

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Notes

  1. 1.

    Readers may find a number of interesting papers on the theory of holomorphic mappings in Chern, Selected Papers (Chern [78]).

  2. 2.

    In general, a function ψ:W→[−∞,∞) defined on an open subset W of R n is said to be subharmonic if ψ is upper semicontinuous and satisfies the submean property in the sense of (2.1.25).

  3. 3.

    K. Oka, in [Iw] VII, [50] and [51], proved three fundamental coherence theorems for (i) 𝒪 Ω , (ii) ℐ〈N〉, and (iii) the normalization of 𝒪 N . Cf. H. Cartan [50] for another proof of the coherence of ℐ〈N〉, and Grauert–Remmert [84]. K. Oka called ℐ〈N〉 the geometric ideal sheaf (l’idéal géométrique de domaines indéterminés). It is interesting to see the comments in Oka [Sp] and Cartan [79].

  4. 4.

    In S. Lang [87] the notation T f,D is used, but this is not proper and mises an essential point: the order function is not dependent on each D, but determined solely by the complete linear system or by its cohomology class, and this is where the First Main Theorem 2.3.31 makes sense.

  5. 5.

    An estimate of this type in several complex variables is found in Kneser [38] without an explicit formula; cf. Noguchi [75].

  6. 6.

    A discussion on the proof of this lemma with Professors Phong and Demailly at Hayama Symposium on Complex Analysis in Several Variables 2002 was very helpful.

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Authors and Affiliations

  1. The University of Tokyo, Tokyo, Japan

    Emeritus Junjiro Noguchi

  2. Tokyo Institute of Technology, Tokyo, Japan

    Emeritus Junjiro Noguchi

  3. Ruhr-University Bochum, Bochum, Germany

    Jörg Winkelmann

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  1. Emeritus Junjiro Noguchi
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  2. Jörg Winkelmann
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Noguchi, J., Winkelmann, J. (2014). The First Main Theorem. In: Nevanlinna Theory in Several Complex Variables and Diophantine Approximation. Grundlehren der mathematischen Wissenschaften, vol 350. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54571-2_2

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