目录
 前辅文
 Preface
 Chapter 1. Preliminaries
  1.1. Frequently used notation and conventions
  1.2. Normed finite-dimensional vector space
  1.3. Lemmas on the length of modules
  1.4. Image of a homomorphism and its determinant
  1.5. Norm of flat and finite homomorphisms
  1.6. Principal divisor and Weil’s reciprocity law
  1.7. Existence of rational sections not passing through given points
  1.8. Graded modules and ample invertible sheaves
  1.9. Several results on separable extensions of the base field
  1.10. Determinant bundle
  1.11. Complex manifold and Hodge theory
  1.12. Connection and curvature
  1.13. Poincaré-Lelong formula
  1.14. C∞ on reduced complex space
 Chapter 2. Geometry of Numbers
  2.1. Convex set and Minkowski’s theorem
  2.2. Polar dual set and Mahler’s inequality
  2.3. The Brunn-Minkowski theorem
  2.4. Estimate of the number of points in a convex lattice
  2.5. Normed finitely generated Z-module
  2.6. λQ and λZ
 Chapter 3. Arakelov Geometry on Arithmetic Curves
  3.1. Orders
  3.2. Arithmetic Chow group over a reduced order
  3.3. Hermitian R-module
  3.4. Arithmetic Riemann-Roch formula on arithmetic curves
  3.5. Effective estimate of the number of small sections
  3.6. Several formulae on arithmetic degree
  3.7. Volume exactness
  3.8. Ample invertible sheaves on arithmetic curves
 Chapter 4. Arakelov Geometry on Arithmetic Surfaces
  4.1. Deligne’s pairing
  4.2. Green functions on Riemann surfaces
  4.3. Arithmetic Chow groups on arithmetic surfaces
  4.4. Intersection theory on arithmetic surfaces
  4.5. Arakelov metric of dualizing sheaf and adjunction formula
  4.6. Determinant bundles for curves
  4.7. Faltings’ Riemann-Roch theorem on arithmetic surfaces
  4.8. Determinant bundle and theta divisor
  4.9. Existence of Faltings’ metric
 Chapter 5. Arakelov Geometry on General Arithmetic Varieties
  5.1. Preliminaries on algebraic geometry and complex geometry
  5.2. Intersection theory of Cartier divisors on excellent schemes
  5.3. Higher dimensional generalization of Weil’s reciprocity law in complex geometry
  5.4. Intersection theory on arithmetic varieties
  5.5. Characteristic classes and Bott-Chern secondary characteristic form
  5.6. Arithmetic characteristic classes
  5.7. Arithmetic Riemann-Roch formula
  5.8. Multi-indexed version of Gromov’s inequality
  5.9. Arithmetic Hilbert-Samuel formula
  5.10. Several kinds of positivity of C∞-hermitian invertible sheaves
  5.11. Estimation of λQ for a normed graded ring
 Chapter 6. Arithmetic Volume Function and Its Continuity
  6.1. Arithmetic volume function
  6.2. Extension of volume function over Q
  6.3. Continuity of volume function
  6.4. Generalized Hodge index theorem
  6.5. Estimate of the number of small sections
 Chapter 7. Nakai-Moishezon Criterion on an Arithmetic Variety
  7.1. Endmorphism N and its basic properties
  7.2. Bounded extension of holomorphic sections
  7.3. Proof of Nakai-Moishezon’s criterion on an arithmetic variety
  7.4. Arithmetic Hilbert-Samuel formula
 Chapter 8. Arithmetic Bogomolov Inequality
  8.1. Semistable locally free coherent sheaves on algebraic curves
  8.2. Hermite-Einstein metric and stability
  8.3. Arithmetic Bogomolov inequality and its proof
 Chapter 9. Lang-Bogomolov Conjecture
  9.1. Height function
  9.2. Height function on abelian variety
  9.3. Equidistribution theorem
  9.4. Cubic metric on complex abelian variety
  9.5. Bogomolov’s conjecture
  9.6. The Lang-Bogomolov Conjecture
  9.7. Small points with respect to a subgroup of finite rank
  9.8. The proof of Theorem 9.24
 Bibliography
 Index
 前辅文
 Preface
 Chapter 1. Preliminaries
  1.1. Frequently used notation and conventions
  1.2. Normed finite-dimensional vector space
  1.3. Lemmas on the length of modules
  1.4. Image of a homomorphism and its determinant
  1.5. Norm of flat and finite homomorphisms
  1.6. Principal divisor and Weil’s reciprocity law
  1.7. Existence of rational sections not passing through given points
  1.8. Graded modules and ample invertible sheaves
  1.9. Several results on separable extensions of the base field
  1.10. Determinant bundle
  1.11. Complex manifold and Hodge theory
  1.12. Connection and curvature
  1.13. Poincaré-Lelong formula
  1.14. C∞ on reduced complex space
 Chapter 2. Geometry of Numbers
  2.1. Convex set and Minkowski’s theorem
  2.2. Polar dual set and Mahler’s inequality
  2.3. The Brunn-Minkowski theorem
  2.4. Estimate of the number of points in a convex lattice
  2.5. Normed finitely generated Z-module
  2.6. λQ and λZ
 Chapter 3. Arakelov Geometry on Arithmetic Curves
  3.1. Orders
  3.2. Arithmetic Chow group over a reduced order
  3.3. Hermitian R-module
  3.4. Arithmetic Riemann-Roch formula on arithmetic curves
  3.5. Effective estimate of the number of small sections
  3.6. Several formulae on arithmetic degree
  3.7. Volume exactness
  3.8. Ample invertible sheaves on arithmetic curves
 Chapter 4. Arakelov Geometry on Arithmetic Surfaces
  4.1. Deligne’s pairing
  4.2. Green functions on Riemann surfaces
  4.3. Arithmetic Chow groups on arithmetic surfaces
  4.4. Intersection theory on arithmetic surfaces
  4.5. Arakelov metric of dualizing sheaf and adjunction formula
  4.6. Determinant bundles for curves
  4.7. Faltings’ Riemann-Roch theorem on arithmetic surfaces
  4.8. Determinant bundle and theta divisor
  4.9. Existence of Faltings’ metric
 Chapter 5. Arakelov Geometry on General Arithmetic Varieties
  5.1. Preliminaries on algebraic geometry and complex geometry
  5.2. Intersection theory of Cartier divisors on excellent schemes
  5.3. Higher dimensional generalization of Weil’s reciprocity law in complex geometry
  5.4. Intersection theory on arithmetic varieties
  5.5. Characteristic classes and Bott-Chern secondary characteristic form
  5.6. Arithmetic characteristic classes
  5.7. Arithmetic Riemann-Roch formula
  5.8. Multi-indexed version of Gromov’s inequality
  5.9. Arithmetic Hilbert-Samuel formula
  5.10. Several kinds of positivity of C∞-hermitian invertible sheaves
  5.11. Estimation of λQ for a normed graded ring
 Chapter 6. Arithmetic Volume Function and Its Continuity
  6.1. Arithmetic volume function
  6.2. Extension of volume function over Q
  6.3. Continuity of volume function
  6.4. Generalized Hodge index theorem
  6.5. Estimate of the number of small sections
 Chapter 7. Nakai-Moishezon Criterion on an Arithmetic Variety
  7.1. Endmorphism N and its basic properties
  7.2. Bounded extension of holomorphic sections
  7.3. Proof of Nakai-Moishezon’s criterion on an arithmetic variety
  7.4. Arithmetic Hilbert-Samuel formula
 Chapter 8. Arithmetic Bogomolov Inequality
  8.1. Semistable locally free coherent sheaves on algebraic curves
  8.2. Hermite-Einstein metric and stability
  8.3. Arithmetic Bogomolov inequality and its proof
 Chapter 9. Lang-Bogomolov Conjecture
  9.1. Height function
  9.2. Height function on abelian variety
  9.3. Equidistribution theorem
  9.4. Cubic metric on complex abelian variety
  9.5. Bogomolov’s conjecture
  9.6. The Lang-Bogomolov Conjecture
  9.7. Small points with respect to a subgroup of finite rank
  9.8. The proof of Theorem 9.24
 Bibliography
 Index