# Arithmetic geometry of toric varieties. metrics, measures and heights

@article{Gil2014ArithmeticGO, title={Arithmetic geometry of toric varieties. metrics, measures and heights}, author={Jos{\'e} I. Burgos Gil and Patrice Philippon and Mart{\'i}n Sombra}, journal={arXiv: Algebraic Geometry}, year={2014} }

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…

## 65 Citations

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Given a toric metrized R-divisor on a toric variety over a global field, we give a formula for the essential minimum of the associated height function. Under suitable positivity conditions, we also…

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We continue with our study of the arithmetic geometry of toric varieties. In this text, we study the positivity properties of metrized R-divisors in the toric setting. For a toric metrized R-divisor,…

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Let X be a normal and geometrically integral projective variety over a global field K and let D be an adelic Cartier divisor on X. We prove a conjecture of Chen, showing that the essential minimum…

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Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L, we show in the following two cases that the semipositive envelope is a…

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