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Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights

José Ignacio Burgos Gil, Patrice Philippon, and Martín Sombra
Société Mathématique de France
Publication Date: 
Number of Pages: 
Astérisque 360
[Reviewed by
Felipe Zaldivar
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Toric varieties are a family of algebraic varieties that are classical, naturally defined, and a natural testing ground for conjectures on algebraic geometry. Toric varieties include affine and projective spaces. Originally defined in Demazure’s paper “Sous-grupes algébriques de rang maximum du group de Cremona” (Ann. Sci. École Norm. Sup, 3 (1970), 507–588) very soon they were at the center of research by virtue of their ubiquity and the variety of techniques that could be used to study them. Moreover, early on their history the didactic value of these varieties when teaching algebraic geometry was recognized. They are easy to study because they are given by ideals generated by certain naturally-occurring monomials. There are now several monographs and textbooks devoted to the toric geometry, from the terse Convex Bodies and Algebraic Geometry monograph by T. Oda (Springer, 1988) to the friendly Toric Varieties graduate textbook by Cox, Little and Schenck (AMS, 2011).

A large part of the ever-growing interest in toric varieties is perhaps due to the rich combinatorics associated, from the very definition, to them. Cones, fans, polytopes, polyhedra, and lattices encode in a natural fashion the definition and properties of toric varieties. Moreover, these close relations allow computations with toric varieties exploiting the convex geometry underlying these varieties.

The monograph under review focuses on the arithmetic side of toric varieties, exploiting the dictionary that translates algebraic geometry properties of toric varieties to combinatorial properties of polytopes and fans.

The arithmetical problem that motivates the monograph is the study of the height of a toric variety. This is a fundamental invariant of a proper variety defined over the rational numbers. The definition of this height is via Arakelov geometry, as developed by Gillet and Soulé after the pioneering work of Arakelov on surfaces. Most of the monograph is devoted to exploring properties of this metric. One of the main results is a closed formula that allows the computation of the height of toric varieties in terms of an integral, over a polytope, of certain concave functions.

Even taking into account that this book is a research monograph, the exposition is systematically clear, including details on the geometry of toric varieties, metrized line bundles, and metrics and measures on toric varieties.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is

  • 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