# Arithmetic Geometry of Toric Varieties: Metrics, Measures and Heights

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*José Ignacio Burgos Gil; Partice Philippon; Martín Sombra*

A publication of the Société Mathématique de France

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.

#### Readership

Graduate students and research mathematicians interested in toric varieties.

#### Reviews & Endorsements

... [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