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Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights
José Ignacio Burgos Gil, Instituto de Ciencias Matemáticas, UAM, Madrid, Spain, Partice Philippon, Institut de Mathématiques de Jussieu, UMR, Paris, France, and Martín Sombra, ICREA and Universitat de Barcelona, Spain
A publication of the Société Mathématique de France.
2014; 222 pp; softcover
Number: 360
ISBN-10: 2-85629-783-8
ISBN-13: 978-2-85629-783-4
List Price: US$78
Member Price: US$62.40
Order Code: AST/360
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The authors 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, the authors study the Arakelov geometry of toric varieties. In particular, they consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. They show that these notions can be translated in terms of convex analysis and are closely related to objects such as polyhedral complexes, concave functions, real Monge-Ampére measures, and Legendre-Fenchel duality.

The authors 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 them to compute the height of toric varieties with respect to some interesting metrics arising from polytopes and compute the height of toric projective curves with respect to the Fubini-Study metric and the height of some toric bundles.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in toric varieties.


"... [T]he exposition is systematically clear, including details on the geometry of toric varieties, metrized line bundles, and metrics and measures on toric varieties."

-- MAA Reviews

Table of Contents

  • Introduction
  • Conventions and notations
  • 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
  • Bibliography
  • List of Symbols
  • Index
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