The First Main Theorem

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.