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Test configurations and Okounkov bodies

Journal article
Authors David Witt Nyström
Published in Compositio Mathematica
Volume 148
Issue 6
Pages 1736-1756
ISSN 0010-437X
Publication year 2012
Published at Department of Mathematical Sciences
Pages 1736-1756
Language en
Links dx.doi.org/10.1112/s0010437x1200035...
https://gup.ub.gu.se/file/98100
Keywords projective manifold, ample line bundle, Okounkov body, test configuration, stability, scalar curvature, linear series, geodesic rays, monge-ampere, stability, varieties, metrics
Subject categories Mathematics

Abstract

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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