Arithmetic geometry of toric varieties
Subtitle: metrics, measures and heights
By: José Ignacio Burgos Gil, Patrice Philippon, Martín Sombra
Contributors: Sombra, Martín, 1970- author; Philippon, Patrice, 1954- author
Series: Astérisque -- 360; Astérisque -- 360.
Publisher: Société Mathématique de France
Published: 2014
Place of publication: Paris
Language: eng
Number of pages: 222
ISBN-13: 9782856297834
Subjects: Toric varieties; Arakelov theory
Description: 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"--Page 4 of cover.
Notes: Includes bibliographical references (pages 207-212) and index.

Abstract also in French.
Table of contents: title: Metrized line bundles and their associated heights; title: The Legendre-Fenchel duality; title: Toric varieties; title: Metrics and measures on toric varieties; title: Height of toric varieties; title: Metrics from polytopes; title: Variations on Fubini-Study metrics.