Journal
COMPOSITIO MATHEMATICA
Volume 148, Issue 6, Pages 1736-1756Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1112/S0010437X12000358
Keywords
projective manifold; ample line bundle; Okounkov body; test configuration; stability
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We associate to a test configuration for a polarized variety a filtration of the section ring of the line bundle. Using the recent work of Boucksom and 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 C-x-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.
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