Test configurations and Okounkov bodies

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.

24 pages

References in corpus (3)

• Okounkov bodies of filtered linear series
• Lectures on Stability and Constant Scalar Curvature
• Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory

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