Verfügbarkeit: Arithmetic geometry of toric varieties

Arithmetic geometry of toric varieties: metrics, measures and heights

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Bibliographische Detailangaben
Beteiligte Personen:	Burgos Gil, José I. 1962- (VerfasserIn), Philippon, Patrice 1954- (VerfasserIn), Sombra, Martín 1970- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Paris Soc. Math. de France 2014
Schriftenreihe:	Astérisque 360
Schlagwörter:	Torische Varietät
Links:	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
Umfang:	VI, 222 S. graph. Darst.
ISBN:	9782856297834

Internformat

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Datensatz im Suchindex

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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.
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author	Burgos Gil, José I. 1962- Philippon, Patrice 1954- Sombra, Martín 1970-
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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
volume_link	(DE-604)BV002579439
work_keys_str_mv	AT burgosgiljosei arithmeticgeometryoftoricvarietiesmetricsmeasuresandheights AT philipponpatrice arithmeticgeometryoftoricvarietiesmetricsmeasuresandheights AT sombramartin arithmeticgeometryoftoricvarietiesmetricsmeasuresandheights

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