Distribution of Values of Holomorphic Mappings (Translations of Mathematical Monographs)

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. Cover......Page 1 Title Page......Page 2 Copyright Page......Page 3 Contents......Page 4 Introduction......Page 8 1. Computation of volumes in C^m......Page 12 2. The homogeneous metric form......Page 18 3. Multiplicity of analytic sets......Page 22 4. The counting function......Page 27 5. The Poincare form......Page 34 2. The characteristic function......Page 36 6. Divisors......Page 37 7. Line bundles......Page 38 8. Hermitian bundles......Page 43 9. The characteristic function......Page 44 10. Higher characteristic functions......Page 49 11. Currents......Page 54 12. The Poincare-Lelong formula......Page 58 13. The relation between characteristic and counting functions......Page 63 1. The case of divisors......Page 68 2. First applications......Page 72 3. The case of sets of codimension greater than 1......Page 75 4. On the Nevanlinna inequality for codimensions greater than 1......Page 77 5. Sokhotskii's theorem for codimensions greater than 1......Page 80 6. Singular volume form......Page 83 7. Preliminary formulation......Page 88 8. Main formulation......Page 92 9. Picard's theorem......Page 96 10. Examples......Page 97 11. Defect relation......Page 99 12. Examples......Page 103 1. Holomorphic curves and their representation......Page 106 2. Grassmann algebra......Page 108 3. Associated curves......Page 110 4. Metric forms......Page 113 5. Characteristic functions......Page 117 6. The case of entire curves......Page 119 9. Second main theorem......Page 120 7. Contact functions......Page 121 8. Two relations......Page 123 9. Second main theorem......Page 127 10. Defect relation and Borel's theorem......Page 132 11. Bin Picard theorem......Page 135 12. More theorems of Picard type......Page 139 1. Exhaustion functions......Page 144 2. Generalization of the main theorems......Page 147 3. The case of holomorphic curves......Page 151 4. The hyperbolic case......Page 158 5. Quadratic transformation......Page 159 6. Singularities of intersection......Page 167 7. Arbitrary singularities......Page 171 1. P-measure......Page 176 2. P-capacity......Page 181 3. Polarity of the set of defective divisors......Page 184 4. On the Bezout problem......Page 188 14. Mappings of finite order......Page 192 5. Estimates of characteristic functions from above......Page 193 6. Mappings with q-regular growth......Page 196 7. Complex variations......Page 203 8. Applications and examples......Page 207 SUPPLEMENT. A Brief Survey of Other Work......Page 212 Bibliography......Page 220 Back Cover......Page 233