Test configurations and Okounkov bodies
Journal article, 2012
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.
geodesic rays
ample line bundle
metrics
projective manifold
Okounkov body
test configuration
varieties
linear series
stability
scalar curvature
stability
monge-ampere
Author
David Witt Nyström
University of Gothenburg
Chalmers, Mathematical Sciences, Mathematics
Compositio Mathematica
0010-437X (ISSN) 1570-5846 (eISSN)
Vol. 148 6 1736-1756Subject Categories
Mathematics
DOI
10.1112/s0010437x12000358