Arithmetic geometry of toric varieties: metrics, measures and heights
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Paris
Soc. Math. de France
2014
|
| Series: | Astérisque
360 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | VI, 222 S. graph. Darst. |
| ISBN: | 9782856297834 |
Staff View
MARC
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Record in the Search Index
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| adam_text | CONTENTS
Introduction ............................................................... 1
Conventions and notations ................................................. 13
1· Metrized line bundles and their associated heights .................... 15
1.1. Smooth metrics in the Archimedean case ........................... 15
1.2. Berkovich spaces of schemes....................................... 18
1.3. Algebraic metrics in the non-Archimedean case .................... 20
1.4. Semipositive and DSP metrics, measures and local heights ......... 27
1.5. Metrics and global heights over adelic fields .................... 32
2. The Legendre-Fenchel duality ........................................... 39
2.1. Convex sets and convex decompositions ............................. 39
2.2. The Legendre-Fenchel dual of a concave function .................. 44
2.3. Operations on concave functions and duality ...................... 49
2.4. The differentiable case ........................................... 54
2.5. The piecewise affine case ........................................ 57
2.6. Differences of concave functions ................................. 66
2.7. Monge-Ampere measures ............................................. 69
3. Toric varieties ........................................................ 77
3.1. Fans and toric varieties ......................................... 77
3.2. Orbits and equi variant morphisms ................................ 79
3.3. T-Cartier divisors and toric line bundles ........................ 84
3.4. Positivity properties of T-Cartier divisors ...................... 90
3.5. Toric schemes over a discrete valuation ring ..................... 94
3.6. T-Cartier divisors on toric schemes ..............................102
3.7. Positivity on toric schemes ......................................107
vi CONTENTS
4. Metrics and measures on toric varieties .................................115
4.1. The variety with corners As(M o) ...................................115
4.2. Analytic torus actions ..............................................120
4.3. Toric metrics ..................................................... 127
4.4. Smooth metrics and their associated measures .......................137
4.5. Algebraic metrics from toric models ................................141
4.6. The one-dimensional case ...........................................145
4.7. Algebraic metrics and their associated measures ....................154
4.8. Semipositive and DSP metrics .......................................157
4.9. Adelic toric metrics ................................................162
5. Height of toric varieties ...............................................165
5.1. Local heights of toric varieties ...................................165
5.2. Global heights of toric varieties ..................................174
6. Metrics from polytopes ................................................... 179
6.1. Integration on polytopes ........................................... . 179
6.2. Metrics and heights from poly topes .................................186
6.3. Heights and entropy .............................................189 7 7
7. Variations on Fubini-Study metrics ....................................193
7.1. Height of toric projective curves ...................................193
7.2. Height of toric bundles .............................................200
Bibliography ...............................................................207
List of Symbols ............................................................213
Index ......................................................................219
We show that the height of a tone variety with respect to a toric
metrized line bundle can be expressed as the integral over a poly-
tope of a certain adelic family of concave functions. To state
and prove this result, we study the Arakelov geometry of toric
varieties. In particular, we consider models over a discrete valua-
tion ring, metrized line bundles, and their associated measures and
heights. We show that these notions can be translated in terms of
convex analysis, and are closely related to objects like polyhedral
complexes, concave functions, real Monge-Ampere measures, and
Legendre-Fenchel duality.
We also present a closed formula for the integral over a polytope
of a function of one variable composed with a linear form. This
formula allows us to compute the height of toric varieties with
respect to some interesting metrics arising from polytopes. We
also compute the height of toric projective curves with respect to
the Fubini-Study metric and the height of some toric bundles.
|
| any_adam_object | 1 |
| author | Burgos Gil, José I. 1962- Philippon, Patrice 1954- Sombra, Martín 1970- |
| author_GND | (DE-588)173531660 (DE-588)1052810772 (DE-588)1052811019 |
| author_facet | Burgos Gil, José I. 1962- Philippon, Patrice 1954- Sombra, Martín 1970- |
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| ctrlnum | (OCoLC)881901765 (DE-599)BVBBV041915589 |
| discipline | Mathematik |
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| id | DE-604.BV041915589 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:08:13Z |
| institution | BVB |
| isbn | 9782856297834 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027359207 |
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| physical | VI, 222 S. graph. Darst. |
| publishDate | 2014 |
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| publisher | Soc. Math. de France |
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| series | Astérisque |
| series2 | Astérisque |
| spelling | Burgos Gil, José I. 1962- Verfasser (DE-588)173531660 aut Arithmetic geometry of toric varieties metrics, measures and heights José Ignacio Burgos Gil ; Patrice Philippon ; Martín Sombra Paris Soc. Math. de France 2014 VI, 222 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Astérisque 360 Torische Varietät (DE-588)4786945-8 gnd rswk-swf Torische Varietät (DE-588)4786945-8 s DE-604 Philippon, Patrice 1954- Verfasser (DE-588)1052810772 aut Sombra, Martín 1970- Verfasser (DE-588)1052811019 aut Astérisque 360 (DE-604)BV002579439 360 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027359207&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027359207&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Burgos Gil, José I. 1962- Philippon, Patrice 1954- Sombra, Martín 1970- Arithmetic geometry of toric varieties metrics, measures and heights Astérisque Torische Varietät (DE-588)4786945-8 gnd |
| subject_GND | (DE-588)4786945-8 |
| title | Arithmetic geometry of toric varieties metrics, measures and heights |
| title_auth | Arithmetic geometry of toric varieties metrics, measures and heights |
| title_exact_search | Arithmetic geometry of toric varieties metrics, measures and heights |
| title_full | Arithmetic geometry of toric varieties metrics, measures and heights José Ignacio Burgos Gil ; Patrice Philippon ; Martín Sombra |
| title_fullStr | Arithmetic geometry of toric varieties metrics, measures and heights José Ignacio Burgos Gil ; Patrice Philippon ; Martín Sombra |
| title_full_unstemmed | Arithmetic geometry of toric varieties metrics, measures and heights José Ignacio Burgos Gil ; Patrice Philippon ; Martín Sombra |
| title_short | Arithmetic geometry of toric varieties |
| title_sort | arithmetic geometry of toric varieties metrics measures and heights |
| title_sub | metrics, measures and heights |
| topic | Torische Varietät (DE-588)4786945-8 gnd |
| topic_facet | Torische Varietät |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027359207&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027359207&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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