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Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties1)

Published online by Cambridge University Press:  22 January 2016

Junjiro Noguchi
Junjiro Noguchi*
Affiliation:
Department of Mathematics, College of General Education, Osaka University
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Nevanlinna’s lemma on logarithmic derivatives played an essential role in the proof of the second main theorem for meromorphic functions on the complex plane C (cf., e.g., [17]). In [19, Lemma 2.3] it was generalized for entire holomorphic curves f: C → M in a compact complex manifold M (Lemma 2.3 in [19] is still valid for non-Kähler M). Here we call, in general, a holomorphic mapping from a domain of C or a Riemann surface into M a holomorphic curve in M, and sometimes use it in the sense of its image if no confusion occurs. Applying the above generalized lemma on logarithmic derivatives to holomorphic curves f: C → V in a complex projective algebraic smooth variety V and making use of Ochiai [22, Theorem A], we had an inequality of the second main theorem type for f and divisors on V (see [19, Main Theorem] and [20]). Other generalizations of Nevanlinna’s lemma on logarithmic derivatives were obtained by Nevanlinna [16], Griffiths-King [10, § 9] and Vitter [23].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 83 , October 1981 , pp. 213 - 233
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

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