Volume arithm\'etique de certains fibr\'es en droites hermitiens sur une vari\'et\'e torique lisse

TL;DR

This paper derives a formula for the arithmetic volume of certain hermitian line bundles on smooth projective toric varieties over Spec(Z), using Fenchel-Legendre transforms related to the metric.

Contribution

It introduces a new explicit formula for the arithmetic volume of equivariant hermitian line bundles on smooth toric varieties, linking it to convex analysis tools.

Findings

Provides a formula for arithmetic volume in terms of Fenchel-Legendre transform.

Connects geometric data of line bundles with convex analysis techniques.

Enhances understanding of arithmetic invariants on toric varieties.

Abstract

Let $X$ be a smooth projective toric variety over $Spec(Z)$. Let $\bar{\mathcal{O}(D)}$ be an equivariant hermitian line bundle on $X$, equipped with a positive metric and invariant under the action of the compact torus of $X(C)$, we provide a formula for the arithmetic volume in term of Fenchel-Legendre transform attached to the metric.