Arithmetic geometry of toric varieties. metrics, measures and heights / José Ignacio Burgos Gil, Patrice Philippon, Martín Sombra.
We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope 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 mode...
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| Main Authors: | , , |
|---|---|
| Language: | English |
| Language and/or Writing System: |
Text in English; abstract also in French. |
| Published: |
Paris :
Société Mathématique de France,
2014.
|
| Series: | Astérisque ;
360. |
| Subjects: | |
| Physical Description: | vi, 222 pages : illustrations ; 24 cm. |
| Format: | Book |
MARC
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| 100 | 1 | |a Burgos Gil, José I. |q (José Ignacio), |d 1962- |e author. |0 http://id.loc.gov/authorities/names/n2001014715 | |
| 245 | 1 | 0 | |a Arithmetic geometry of toric varieties. metrics, measures and heights / |c José Ignacio Burgos Gil, Patrice Philippon, Martín Sombra. |
| 264 | 1 | |a Paris : |b Société Mathématique de France, |c 2014. | |
| 300 | |a vi, 222 pages : |b illustrations ; |c 24 cm. | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a unmediated |b n |2 rdamedia | ||
| 338 | |a volume |b nc |2 rdacarrier | ||
| 490 | 1 | |a Astérisque, |x 0303-1179 ; |v 360 | |
| 504 | |a Includes bibliographical references (pages [207]-212) and index. | ||
| 505 | 0 | |a Metrized line bundles and their associated heights -- The Legendre-Fenchel duality -- Toric varieties -- Metrics and measures on toric varieties -- Height of toric varieties -- Metrics from polytopes -- Variations on Fubini-Study metrics. | |
| 520 | 3 | |a We show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope 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 valuation 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-Ampère 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."--Back cover. | |
| 546 | |a Text in English; abstract also in French. | ||
| 650 | 0 | |a Toric varieties. |0 http://id.loc.gov/authorities/subjects/sh93002816 | |
| 650 | 0 | |a Arakelov theory. |0 http://id.loc.gov/authorities/subjects/sh00000203 | |
| 700 | 1 | |a Sombra, Martín, |d 1970- |e author. |0 http://id.loc.gov/authorities/names/no2014077804 | |
| 700 | 1 | |a Philippon, Patrice, |d 1954- |e author. |0 http://id.loc.gov/authorities/names/n92018229 | |
| 830 | 0 | |a Astérisque ; |v 360. |0 http://id.loc.gov/authorities/names/n86716119 | |
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