Value Distribution Theory for Meromorphic Maps
Value distribution theory studies the behavior of mermorphic maps. Let f: M - N be a merom orphic map between complex manifolds. A target family CI ~ (Ea1aEA of analytic subsets Ea of N is given where A is a connected. compact complex manifold. The behavi
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Value Distribution Theory for Meromorphic Maps
Aspects of Mathematics Aspekte der Mathematik Editor: Klas Diederich
Vol. E1: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part A Vol. E2: M. Knebusch/M. Kolster, Wittrings Vol. E3: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part B Vol. E4: M. Laska, Elliptic qurves over Number Fields with Prescribed Reduction Type Vol. E5: P. Stiller, Auto""orphic Forms and the Picard Number of an Elliptic Surface Vol. E6: G. Faltings, G. Wustholz et aI., Rational Points (A Pubiication of the Max·Planck-lnstitut fur Mathematik, Bonn)
Vol. E7: W. Stoll, Value Distribution Theory for Meromorphic Maps Vol. D1: H. Kraft, Geometrische Methoden in der I nvariantentheorie
The texts published in this series are intended for graduate students and all mathematicians who wish to broaden their research horizons or who simply want to get a better idea of what is going on in a given field. They are introductions to areas close to modern research at a high level and prepare the reader for a better understanding of research papers. Many of the books can also be used to supplement graduate course programs. The series comprises two sub-series, one with English texts only and the other in German.
Wilhelm Stoll
Value Distribution Theory for Meromorphic Maps
Springer Fachmedien Wiesbaden GmbH
Prof. Dr. Wilhelm Sto/1 is Professor of Mathematics at the University of Notre Dame, Notre Dame,.lnäiana 46556, USA
AMS Subject Classification: 32 H 30, 32 A 22
ISBN 978-3-663-05294-4 ISBN 978-3-663-05292-0 (eBook) DOI 10.1007/978-3-663-05292-0 1985 All rights reserved
© Springer Fachmedien Wiesbaden 1985
Originally published by Friedr. Vieweg & Sohn Verlagsgesellschaft in 1985. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. without prior permission of the copyright holder. Produced by Lengarieher Handelsdruckerei, Langerich
Dedicated to the memory of Yozo Matsushima
CONTENTS Page Preface
VIII
Letters
XI
Introduction
1
A.
Value Distribution Theory for Fixed Targets
1
B.
Value Distribution Theory for Moving Targets
56
1.
Hermitian Geometry
92
2.
Meromorphic Maps on Parabolic Manifolds
115
3.
The First Main Theorem
134
4.
Associated Maps
151
5.
Frenet Frames
163
6.
The Ahlfors Estimates
191
7.
General Position
216
8.
The Second Main Theorem
245
9.
Value Distribution over a Function Field
275
10.
An Example
310
11.
The Theorem of Nevanlinna-Mori
317
12.
References
334
13.
Index
344
Preface Value distribution theory studies the behavior of mermorphic maps. Let
f: M -
N
target family
CI
be a merom orphic map between complex manifolds. ~
(Ea1aEA
of analytic subsets
A is a connected. compact complex manifold. family
["'(CI)
(f- 1{E a )laEA
=
created by many contributors.
is investigated.
Ea
of
N
The behavior of the inverse A substantial theory has been
Usually th
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