# Distribution of Values of Holomorphic Mappings

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*B. V. Shabat*

A vast literature has grown up around the value distribution theory of
meromorphic functions, synthesized by Rolf Nevanlinna in the 1920s and
singled out by Hermann Weyl as one of the greatest mathematical
achievements of this century. The multidimensional aspect, involving
the distribution of inverse images of analytic sets under holomorphic
mappings of complex manifolds, has not been fully treated in the
literature. This volume thus provides a valuable introduction to
multivariate value distribution theory and a survey of some of its
results, rich in relations to both algebraic and differential geometry
and surely one of the most important branches of the modern geometric
theory of functions of a complex variable.

Since the book begins with preparatory material from the contemporary
geometric theory of functions, only a familiarity with the elements of
multidimensional complex analysis is necessary background to understand
the topic. After proving the two main theorems of value distribution
theory, the author goes on to investigate further the theory of
holomorphic curves and to provide generalizations and applications of
the main theorems, focusing chiefly on the work of Soviet
mathematicians.

#### Table of Contents

# Table of Contents

## Distribution of Values of Holomorphic Mappings

- Cover Cover11
- Title page i2
- Contents iii4
- Introduction 18
- The Characteristic and Counting Functions 512
- The Main Theorems of Value Distribution Theory 6168
- Holomorphic Curves 99106
- Generalization of the Main Theorems 137144
- Further Results 169176
- Supplement. A Brief Survey of Other Work 205212
- Bibliography 213220
- Back Cover Back Cover1233

- Cover Cover1234
- Title page i235
- Contents iii237
- Introduction 1241
- The Characteristic and Counting Functions 5245
- The Main Theorems of Value Distribution Theory 61301
- Holomorphic Curves 99339
- Generalization of the Main Theorems 137377
- Further Results 169409
- Supplement. A Brief Survey of Other Work 205445
- Bibliography 213453
- Back Cover Back Cover1466

- Cover Cover1467
- Title page i468
- Contents iii470
- Introduction 1474
- The Characteristic and Counting Functions 5478
- The Main Theorems of Value Distribution Theory 61534
- Holomorphic Curves 99572
- Generalization of the Main Theorems 137610
- Further Results 169642
- Supplement. A Brief Survey of Other Work 205678
- Bibliography 213686
- Back Cover Back Cover1699