Arakelov geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2014
|
| Series: | Translations of mathematical monographs
244 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027889556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Originally published in Japanese: Arakerofu kika (Tokyo : Iwanam Shoten, 2008). - Includes bibliographical references and index |
| Physical Description: | IX, 285 S. |
| ISBN: | 9781470410742 |
Staff View
MARC
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| 490 | 1 | |a Translations of mathematical monographs |v 244 | |
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| 650 | 4 | |a Arakelov theory | |
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Record in the Search Index
| _version_ | 1819290650828341248 |
|---|---|
| adam_text | Contents
Preface vii
Chapter 1. Preliminaries 1
1.1. Frequently used notation and conventions 1
1.2. Normed finite-dimensional vector space 1
1.3. Lemmas on the length of modules 6
1.4. Image of a homomorphism and its determinant 11
1.5. Norm of fiat and finite homomorphisms 15
1.6. Principal divisor and Weil’s reciprocity law 17
1.7. Existence of rational sections not passing through given points 22
1.8. Graded modules and ample invertible sheaves 24
1.9. Several results on separable extensions of the base field 26
1.10. Determinant bundle 28
1.11. Complex manifold and Hodge theory 31
1.12. Connection and curvature 35
1.13. Poincare-Lelong formula 36
1.14. C°° on reduced complex space 39
Chapter 2. Geometry of Numbers 41
2.1. Convex set and Minkowski s theorem 41
2.2. Polar dual set and Mahler’s inequality 44
2.3. The Brunn-Minkowski theorem 45
2.4. Estimate of the number of points in a convex lattice 48
2.5. Normed finitely generated Z-module 55
2.6. Aq and Az 58
Chapter 3. Arakelov Geometry on Arithmetic Curves 63
3.1. Orders 63
3.2. Arithmetic Chow group over a reduced order 69
3.3. Hermitian R-module 70
3.4. Arithmetic Riemann-Roch formula on arithmetic curves 74
3.5. Effective estimate of the number of small sections 77
3.6. Several formulae on arithmetic degree 80
3.7. Volume exactness 83
3.8. Ample invertible sheaves on arithmetic curves 84
Chapter 4. Arakelov Geometry on Arithmetic Surfaces 87
4.1. Deligne s pairing 87
4.2. Green functions on Riemann surfaces 95
4.3. Arithmetic Chow groups on arithmetic surfaces 101
iii
IV
CONTENTS
4.4. Intersection theory on arithmetic surfaces 104
4.5. Arakelov metric of dualizing sheaf and adjunction formula 110
4.6. Determinant bundles for curves 114
4.7. Fallings’ Riemann֊Roch theorem on arithmetic surfaces 119
4.8. Determinant bundle and theta divisor 124
4.9. Existence of Fairings’ metric 129
Chapter 5. Arakelov Geometry on General Arithmetic Varieties 141
5.1. Preliminaries on algebraic geometry and complex geometry 141
5.2. Intersection theory of Cartier divisors on excellent schemes 146
5.3. Higher dimensional generalization of Weil’s reciprocity law
in complex geometry 150
5.4. Intersection theory on arithmetic varieties 155
5.5. Characteristic classes and Bott-Chern secondary characteristic form 166
5.6. Arithmetic characteristic classes 171
5.7. Arithmetic Riemann-Roch formula 174
5.8. Multi-indexed version of Gromov’s inequality 175
5.9. Arithmetic Hilbert-Samuel formula 180
5.10. Several kinds of positivity of C^-hermitian invertible sheaves 185
5.11. Estimation of Xq for a normed graded ring 190
Chapter 6. Arithmetic Volume Function and Its Continuity 197
6.1. Arithmetic volume function 197
6.2. Extension of volume function over Q 202
6.3. Continuity of volume function 204
6.4. Generalized Hodge index theorem 208
6.5. Estimate of the number of small sections 212
Chapter 7. Nakai-Moishezon Criterion on an Arithmetic Variety 225
7.1. Endmorphism N and its basic properties 225
7.2. Bounded extension of holomorphic sections 228
7.3. Proof of Nakai-Moishezon’s criterion on an arithmetic variety 234
7.4. Arithmetic Hilbert-Samuel formula 235
Chapter 8. Arithmetic Bogomolov Inequality 237
8.1. Semistable locally free coherent sheaves on algebraic curves 237
8.2. H ermite- Einstein metric and stability 239
8.3. Arithmetic Bogomolov inequality and its proof 243
Chapter 9. Lang-Bogomolov Conjecture 249
9.1. Height function 249
9.2. Height function on abelian variety 256
9.3. Equidistribution theorem 259
9.4. Cubic metric on complex abelian variety 263
9.5. Bogomolov s conjecture 265
9.6. The Lang-Bogomolov Conjecture 269
9.7. Small points with respect to a subgroup of finite rank 270
9.8. The proof of Theorem 9.24 275
Bibliography
Index
CONTENTS
V
279
283
|
| any_adam_object | 1 |
| author | Moriwaki, Atsushi 1960- |
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| author_facet | Moriwaki, Atsushi 1960- |
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| dewey-full | 516.3/5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
| dewey-search | 516.3/5 |
| dewey-sort | 3516.3 15 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042454328 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T17:11:43Z |
| institution | BVB |
| isbn | 9781470410742 |
| language | English |
| lccn | 014021155 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027889556 |
| oclc_num | 903305999 |
| open_access_boolean | |
| owner | DE-11 DE-355 DE-BY-UBR |
| owner_facet | DE-11 DE-355 DE-BY-UBR |
| physical | IX, 285 S. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spellingShingle | Moriwaki, Atsushi 1960- Arakelov geometry Translations of mathematical monographs Arakelov theory Geometry, Algebraic |
| title | Arakelov geometry |
| title_alt | Arakerofu kika |
| title_auth | Arakelov geometry |
| title_exact_search | Arakelov geometry |
| title_full | Arakelov geometry Atsushi Moriwaki, translated by the author |
| title_fullStr | Arakelov geometry Atsushi Moriwaki, translated by the author |
| title_full_unstemmed | Arakelov geometry Atsushi Moriwaki, translated by the author |
| title_short | Arakelov geometry |
| title_sort | arakelov geometry |
| topic | Arakelov theory Geometry, Algebraic |
| topic_facet | Arakelov theory Geometry, Algebraic |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027889556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
| work_keys_str_mv | AT moriwakiatsushi arakerofukika AT moriwakiatsushi arakelovgeometry |