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Test configurations and Okounkov bodies

Part of: Families, fibrations Deformations of analytic structures Complex manifolds

Published online by Cambridge University Press:  11 October 2012

David Witt Nyström
David Witt Nyström*
Affiliation:
Chalmers University of Technology and University of Göteborg, Göteborg, Sweden (email: [email protected])
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Abstract

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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×-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

Keywords

projective manifoldample line bundleOkounkov bodytest configurationstability

MSC classification

Secondary: 14D20: Algebraic moduli problems, moduli of vector bundles 32G08: Deformations of fiber bundles 32Q15: Kähler manifolds 32Q26: Notions of stability
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 6 , November 2012 , pp. 1736 - 1756
Copyright
Copyright © Foundation Compositio Mathematica 2012

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