Test configurations and Okounkov bodies
Artikel i vetenskaplig tidskrift, 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

Författare

David Witt Nyström

Göteborgs universitet

Chalmers, Matematiska vetenskaper, Matematik

Compositio Mathematica

0010-437X (ISSN) 1570-5846 (eISSN)

Vol. 148 6 1736-1756

Ämneskategorier

Matematik

DOI

10.1112/s0010437x12000358