Arithmetic geometry of toric varieties. Metrics, measures and heights

TL;DR

This paper expresses the height of toric varieties in terms of integrals over polytopes involving concave functions, linking Arakelov geometry with convex analysis to facilitate explicit height computations.

Contribution

It provides a new integral formula for heights of toric varieties using convex analysis and Legendre-Fenchel duality, connecting Arakelov geometry with polyhedral geometry.

Findings

Height expressed as polytope integral of concave functions

Closed formula for integrals of composed functions over polytopes

Explicit height calculations for toric curves and bundles

Abstract

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-Amp\`ere 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 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 of some toric bundles.