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Test configurations and Okounkov bodies

David Witt Nyström (Institutionen för matematiska vetenskaper, matematik)
Compositio Mathematica (0010-437X). Vol. 148 (2012), 6, p. 1736-1756.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: projective manifold, ample line bundle, Okounkov body, test configuration, stability, scalar curvature, linear series, geodesic rays, monge-ampere, stability, varieties, metrics

Denna post skapades 2013-01-03. Senast ändrad 2017-07-03.
CPL Pubid: 168962


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Institutionen för matematiska vetenskaper, matematik (2005-2016)



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