Test configurations and Okounkov bodies

TL;DR

This paper links test configurations of line bundles to Okounkov bodies, generalizing toric geometry results and relating measures on these bodies to deformation and symmetry properties of the manifold.

Contribution

It introduces a filtration-based approach to associate concave functions on Okounkov bodies with test configurations, extending known results in toric geometry.

Findings

Law of the concave function determines asymptotic weight distribution.

Pushforward measure on Okounkov body equals a Duistermaat-Heckman measure.

In special cases, the measure is piecewise polynomial.

Abstract

We associate to a test configuration of an ample line bundle a filtration of the section ring of the line bundle. Using the recent work of Boucksom-Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat-Heckman measure of a certain deformation of the manifold. Via the Duisteraat-Heckman formula, we get as a corollary that in the special case of an effective $\mathbb{C^{\times}}$-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.