Topics in Nevanlinna Theory
These are notes of lectures on Nevanlinna theory, in the classical case of meromorphic functions, and the generalization by Carlson-Griffith to equidimensional holomorphic maps using as domain space finite coverings of C resp. Cn. Conjecturally best possi
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Serg1e Lang
William Cherry
Topics in NevanlinnaTheory
Spnnqer-venaq Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Authors
Serge Lang William Cherry Department of Mathematics, Yale University Box 2155 Yale Station, New Haven, CT 06520, USA
Mathematics Subject Classification (1980): 30035, 32A22, 32H30 ISBN 3-540-52785-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-52785-0 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1990 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
PART ONE LECTURES ON NEVANLINNA THEORY by Serge Lang CHAPTER I NEVANLINNA THEORY IN ONE VARIABLE 1. The Poisson-Jensen formula and the Nevanlinna functions
2. The differential geometric definitions and Green-Jensen's formula 3. Some calculus lemmas 4. Ramification and second main theorem 5. An estimate for the height transform 6. Variations and applications, the lemma on the logarithmic derivative
Appendix by Zhuan Yeo On Nevanlinna's error term
12 20 29 34 42 48
53
CHAPTER II EQUIDIMENSIONAL HIGHER DIMENSIONAL THEORY 1. 2. 3. 4. 5. 6. 7. 8.
The Chern and Ricci forms 57 n 1 Some forms on C" and p - ( C ) and the Green-Jensen formula 66 Stokes' theorem with certain singularities on C" 70 The Nevanlinna functions and the first main theorem 78 The calculus lemma 85 The trace and determinant in the main theorem 87 91 A general second main theorem (Ahlfors-Wong method) Variations and applications 102 1
PART TWO NEVANLINNA THEORY OF COVERINGS by William Cherry
CHAPTER III NEVANLINNA THEORY FOR MEROMORPHIC FUNCTIONS ON COVERINGS OF C 1. Notation and preliminaries
2. 3. 4. 5.
First main theorem Calculus lemmas Ramification and the second main theorem A general second main theorem
113 121 126 128 131
CHAPTER IV EQUIDIMENSIONAL NEVANLINNA THEORY ON COVERINGS OF c-
First main theorem Calculus lemmas The second main theorem without a divisor A general second main theorem A variation
143 151 154 156 158 167
References
169
INDEX
173
1. Notation and preliminaries
2. 3. 4. 5. 6.
2
CHAPTER I NEVANLINNA THEORY IN ONE VARIABLE In the first part of this chapter we essentially follow Nevanlinna, as in his book [Ne]. The main difference lies in the fact that we are careful about the error term in Nevanlinna's main theorem. That this error term has an interesting structure was first brought up in [La 8], in analogy with a similar conject
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