Adelic Divisors on Arithmetic Varieties.
Moriwaki, Atsushi.
Adelic Divisors on Arithmetic Varieties. - 1st ed. - 1 online resource (134 pages) - Memoirs of the American Mathematical Society Series ; v.242 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Introduction -- 0.1. Birational Arakelov geometry -- 0.2. Green functions on analytic spaces over a compete discrete valuation field -- 0.3. Adelic arithmetic divisors -- 0.4. Main results -- 0.5. Conventions and terminology -- Chapter 1. Preliminaries -- 1.1. Lemmas -- 1.2. ℝ-Cartier divisors on a noetherian integral scheme -- 1.3. Analytification of algebraic schemes over a complete valuation field -- 1.4. Green functions on complex varieties -- Chapter 2. Adelic ℝ-Cartier Divisors over a Discrete Valuation Field -- 2.1. Green functions on analytic spaces over a discrete valuation field -- 2.2. Definition of adelic ℝ-Cartier divisors -- 2.3. Local degree -- 2.4. Local intersection number -- Chapter 3. Local and Global Density Theorems -- 3.1. Vertical fractional ideal sheaves and birational system of models -- 3.2. Model functions -- 3.3. Density theorems -- 3.4. Zariski's lemma for integrable functions -- 3.5. Radon measure arising from local intersection number -- Chapter 4. Adelic Arithmetic ℝ-Cartier Divisors -- 4.1. Definition and basic properties -- 4.2. Global degree -- 4.3. Volume of adelic arithmetic ℝ-Cartier divisors -- 4.4. Positivity of adelic arithmetic ℝ-Cartier divisors -- 4.5. Global intersection number -- Chapter 5. Continuity of the Volume Function -- 5.1. Basic properties of the volume -- 5.2. Proof of the continuity of the volume function -- 5.3. Applications -- Chapter 6. Zariski Decompositions of Adelic Arithmetic Divisors on Arithmetic Surfaces -- 6.1. Local Zariski decompositions of adelic divisors on algebraic curves -- 6.2. Proof of Zariski decompositions for adelic arithmetic divisors -- Chapter 7. Characterization of Nef Adelic Arithmetic Divisors on Arithmetic Surfaces -- 7.1. Hodge index theorem for adelic arithmetic divisors -- 7.2. Arithmetic asymptotic multiplicity. 7.3. Necessary condition for the equality ̂ =̂ ᵪ -- 7.4. Numerical characterization -- Chapter 8. Dirichlet's unit Theorem for Adelic Arithmetic Divisors -- 8.1. Fundamental question for adelic divisors -- 8.2. Proof of Theorem 8.1.2 -- Appendix A. Characterization of Relatively Nef Cartier Divisors -- A.1. Asymptotic multiplicity -- A.2. Sectional decomposition -- A.3. Characterization in terms of _ -- Bibliography -- Subject Index -- Symbol Index -- Back Cover.
In this article, the author generalizes several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors, Zariski decompositions for arithmetic divisors on arithmetic surfaces and a special case of Dirichlet's unit theorem on arithmetic varieties, to the case of the adelic arithmetic divisors.
9781470429423
Divisor theory.
Topological groups.
Algebraic varieties.
Approximation theory.
Electronic books.
QA242 .M86 2016
512.7/4
Adelic Divisors on Arithmetic Varieties. - 1st ed. - 1 online resource (134 pages) - Memoirs of the American Mathematical Society Series ; v.242 . - Memoirs of the American Mathematical Society Series .
Cover -- Title page -- Introduction -- 0.1. Birational Arakelov geometry -- 0.2. Green functions on analytic spaces over a compete discrete valuation field -- 0.3. Adelic arithmetic divisors -- 0.4. Main results -- 0.5. Conventions and terminology -- Chapter 1. Preliminaries -- 1.1. Lemmas -- 1.2. ℝ-Cartier divisors on a noetherian integral scheme -- 1.3. Analytification of algebraic schemes over a complete valuation field -- 1.4. Green functions on complex varieties -- Chapter 2. Adelic ℝ-Cartier Divisors over a Discrete Valuation Field -- 2.1. Green functions on analytic spaces over a discrete valuation field -- 2.2. Definition of adelic ℝ-Cartier divisors -- 2.3. Local degree -- 2.4. Local intersection number -- Chapter 3. Local and Global Density Theorems -- 3.1. Vertical fractional ideal sheaves and birational system of models -- 3.2. Model functions -- 3.3. Density theorems -- 3.4. Zariski's lemma for integrable functions -- 3.5. Radon measure arising from local intersection number -- Chapter 4. Adelic Arithmetic ℝ-Cartier Divisors -- 4.1. Definition and basic properties -- 4.2. Global degree -- 4.3. Volume of adelic arithmetic ℝ-Cartier divisors -- 4.4. Positivity of adelic arithmetic ℝ-Cartier divisors -- 4.5. Global intersection number -- Chapter 5. Continuity of the Volume Function -- 5.1. Basic properties of the volume -- 5.2. Proof of the continuity of the volume function -- 5.3. Applications -- Chapter 6. Zariski Decompositions of Adelic Arithmetic Divisors on Arithmetic Surfaces -- 6.1. Local Zariski decompositions of adelic divisors on algebraic curves -- 6.2. Proof of Zariski decompositions for adelic arithmetic divisors -- Chapter 7. Characterization of Nef Adelic Arithmetic Divisors on Arithmetic Surfaces -- 7.1. Hodge index theorem for adelic arithmetic divisors -- 7.2. Arithmetic asymptotic multiplicity. 7.3. Necessary condition for the equality ̂ =̂ ᵪ -- 7.4. Numerical characterization -- Chapter 8. Dirichlet's unit Theorem for Adelic Arithmetic Divisors -- 8.1. Fundamental question for adelic divisors -- 8.2. Proof of Theorem 8.1.2 -- Appendix A. Characterization of Relatively Nef Cartier Divisors -- A.1. Asymptotic multiplicity -- A.2. Sectional decomposition -- A.3. Characterization in terms of _ -- Bibliography -- Subject Index -- Symbol Index -- Back Cover.
In this article, the author generalizes several fundamental results for arithmetic divisors, such as the continuity of the volume function, the generalized Hodge index theorem, Fujita's approximation theorem for arithmetic divisors, Zariski decompositions for arithmetic divisors on arithmetic surfaces and a special case of Dirichlet's unit theorem on arithmetic varieties, to the case of the adelic arithmetic divisors.
9781470429423
Divisor theory.
Topological groups.
Algebraic varieties.
Approximation theory.
Electronic books.
QA242 .M86 2016
512.7/4