Transforming metrics on a line bundle to the Okounkov body

TL;DR

This paper introduces a method to associate convex functions called Chebyshev transforms to metrics on line bundles, linking metric volumes to integrals over Okounkov bodies, and generalizing classical results in complex geometry.

Contribution

It defines the Chebyshev transform for metrics on line bundles and relates metric volume differences to integrals over Okounkov bodies, extending previous theories.

Findings

The Chebyshev transform links metrics to convex geometry.

Metric volume difference equals integral of Chebyshev transform differences.

Proves differentiability of metric volume in the cone of big metrized divisors.

Abstract

Let $L$ be a big holomorphic line bundle on a complex projective manifold $X.$ We show how to associate a convex function on the Okounkov body of $L$ to any continuous metric $\psi$ on $L.$ We will call this the Chebyshev transform of $\psi,$ denoted by $c[\psi].$ Our main theorem states that the difference of metric volume of $L$ with respect to two metrics, a notion introduced by Berman-Boucksom, is equal to the integral over the Okounkov body of the difference of the Chebyshev transforms of the metrics. When the metrics have positive curvature the metric volume coincides with the Monge-Amp\`ere energy, which is a well-known functional in K\"ahler-Einstein geometry and Arakelov geometry. We show that this can be seen as a generalization of classical results on Chebyshev constants and the Legendre transform of invariant metrics on toric manifolds. As an application we prove the differentiability of the metric volume in the cone of big metrized $\mathbb{R}$-divisors. This generalizes the result of Boucksom-Favre-Jonsson on the differentiability of the ordinary volume of big $\mathbb{R}$-divisors and the result of Berman-Boucksom on the differentiability of the metric volume when the underlying line bundle is fixed.