Arakelov Geometry (Translations of Mathematical Monographs)

The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert-Samuel formula, arithmetic Nakai-Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang-Bogomolov conjecture and so on. In addition, the author presents, with full details, the proof of Faltings’ Riemann-Roch theorem. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes. Title 3 Copyright 4 Contents 5 Preface 9 Chapter 1. Preliminaries 13 1.1. Frequently used notation and conventions 13 1.2. Normed finite-dimensional vector space 13 1.3. Lemmas on the length of modules 18 1.4. Image of a homomorphism and its determinant 23 1.5. Norm of flat and finite homomorphisms 27 1.6. Principal divisor and Weil's reciprocity law 29 1.7. Existence of rational sections not passing through given 34 1.8. Graded modules and ample invertible sheaves 36 1.9. Several results on separable extensions of the base field 38 1.10. Determinant bundle 40 1.11. Complex manifold and Hodge theory 43 1.12. Connection and curvature 47 1.13. Poincare-Lelong formula 48 1.14. C°° on reduced complex space 51 Chapter 2. Geometry of Numbers 53 2.1. Convex set and Minkowski's theorem 53 2.2. Polar dual set and Mahler's inequality 56 2.3. The Brunn-Minkowski theorem 57 2.4. Estimate of the number of points in a convex lattice 60 2.5. Normed finitely generated Z-module 67 2.6. Aq and 70 Chapter 3. Arakelov Geometry on Arithmetic Curves 75 3.1. Orders 75 3.2. Arithmetic Chow group over a reduced order 81 3.3. Hermitian i?-module 82 3.4. Arithmetic Riemann-Roch formula on arithmetic curves 86 3.5. Effective estimate of the number of small sections 89 3.6. Several formulae on arithmetic degree 92 3.7. Volume exactness 95 3.8. Ample invertible sheaves on arithmetic curves 96 Chapter 4. Arakelov Geometry on Arithmetic Surfaces 99 4.1. Deligne's pairing 99 4.2. Green functions on Riemann surfaces 107 4.3. Arithmetic Chow groups on arithmetic surfaces 113 4.4. Intersection theory on arithmetic surfaces 116 4.5. Arakelov metric of dualizing sheaf and adjunction formula 122 4.6. Determinant bundles for curves 126 4.7. Faltings' Riemann-Roch theorem on arithmetic surfaces 131 4.8. Determinant bundle and theta divisor 136 4.9. Existence of Faltings' metric 141 Chapter 5. Arakelov Geometry on General Arithmetic Varieties 153 5.1. Preliminaries on algebraic geometry and complex geometry 153 5.2. Intersection theory of Cartier divisors on excellent schemes 158 5.3. Higher dimensional generalization of Weil's reciprocity law in complex geometry 162 5.4. Intersection theory on arithmetic varieties 167 5.5. Characteristic classes and Bott-Chern secondary characteristic form 178 5.6. Arithmetic characteristic classes 183 5.7. Arithmetic Riemann-Roch formula 186 5.8. Multi-indexed version of Gromov's inequality 187 5.9. Arithmetic Hilbert-Samuel formula 192 5.10. Several kinds of positivity of C°°-hermitian invertible sheaves 197 5.11. Estimation of Xq for a normed graded ring 202 Chapter 6. Arithmetic Volume Function and Its Continuity 209 6.1. Arithmetic volume function 209 6.2. Extension of volume function over Q 214 6.3. Continuity of volume function 216 6.4. Generalized Hodge index theorem 220 6.5. Estimate of the number of small sections 224 Chapter 7. Nakai-Moishezon Criterion on an Arithmetic Variety 237 7.1. Endmorphism N and its basic properties 237 7.2. Bounded extension of holomorphic sections 240 7.3. Proof of Nakai-Moishezon's criterion on an arithmetic variety 246 7.4. Arithmetic Hilbert-Samuel formula 247 Chapter 8. Arithmetic Bogomolov Inequality 249 8.1. Semistable locally free coherent sheaves on algebraic curves 249 8.2. Hermite-Einstein metric and stability 251 8.3. Arithmetic Bogomolov inequality and its proof 255 Chapter 9. Lang-Bogomolov Conjecture 261 9.1. Height function 261 9.2. Height function on abelian variety 268 9.3. Equidistribution theorem 271 9.4. Cubic metric on complex abelian variety 275 9.5. Bogomolov's conjecture 277 9.6. The Lang-Bogomolov Conjecture 281 9.7. Small points with respect to a subgroup of finite rank 282 9.8. The proof of Theorem 9.24 287 Bibliography 291 Index 295