Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T00:35:43.483Z Has data issue: false hasContentIssue false
  • English
  • Français

Test configurations and Okounkov bodies

Part of: Deformations of analytic structures Families, fibrations 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: wittnyst@chalmers.se)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

[Ber09]Berndtsson, B., Probability measures associated to geodesics in the space of Kähler metrics, Preprint (2009), math.DG/0907.1806.Google Scholar
[BC11]Boucksom, S. and Chen, H., Okounkov bodies of filtered linear series, Compositio Math. 147 (2011), 1205–1229.CrossRefGoogle Scholar
[BG81]Boutet de Monvel, L. and Guillemin, V., The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99 (Princeton University Press, Princeton, NJ, 1981).Google Scholar
[Don01]Donaldson, S. K., Scalar curvature and projective embeddings. I., J. Differential Geom. 59 (2001), 479–522.CrossRefGoogle Scholar
[Don02]Donaldson, S. K., Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289–349.CrossRefGoogle Scholar
[Don05]Donaldson, S. K., Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), 453–472.CrossRefGoogle Scholar
[DH82]Duistermaat, J. J. and Heckman, G. J., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259–268.CrossRefGoogle Scholar
[KK08]Kaveh, K. and Khovanskii, A., Algebraic equations and convex bodies, in Perspectives in Analysis, Topology and Geometry (in honor of Oleg Viro), Progr. Math., to appear, Preprint (2008), math.AG/0812.4688v1.Google Scholar
[KK09]Kaveh, K. and Khovanskii, A., Newton convex bodies, semigroups of integral points, graded algebras and intersection theory, Preprint (2009), math.AG/0904.3350v2.Google Scholar
[KK10]Kaveh, K. and Khovanskii, A., Convex bodies associated to actions of reductive groups, Preprint (2010), math.AG/1001.4830v1.Google Scholar
[Laz04]Lazarsfeld, R., Positivity in algebraic geometry. I. Classical setting: line bundles and linear series (Springer, Berlin, 2004).Google Scholar
[LM09]Lazarsfeld, R. and Mustaţă, M., Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 783–835.CrossRefGoogle Scholar
[Mab86]Mabuchi, T., K-energy maps integrating Futaki invariants, Tohoko Math. J. (2) 38 (1986), 575–593.Google Scholar
[Oko96]Okounkov, A., Brunn–Minkowski inequality for multiplicities, Invent. Math. 125 (1996), 405–411.CrossRefGoogle Scholar
[PS07]Phong, D. H. and Sturm, J., Test configurations for K-stability and geodesic rays, J. Symplectic Geom. 5 (2007), 221–247.CrossRefGoogle Scholar
[PS08]Phong, D. H. and Sturm, J., Lectures on stability and constant scalar curvature, Preprint (2008), math.DG/0801.4179.Google Scholar
[PS10]Phong, D. H. and Sturm, J., Regularity of geodesic rays and Monge–Ampere equations, Proc. Amer. Math. Soc. 138 (2010), 3637–3650.CrossRefGoogle Scholar
[RT06]Ross, J. and Thomas, R., An obstruction to the existence of constant scalar curvature Kähler metrics, J. Differential Geom. 72 (2006), 429–466.CrossRefGoogle Scholar
[RT07]Ross, J. and Thomas, R., A study of the Hilbert–Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), 201–255.CrossRefGoogle Scholar
[Sem92]Semmes, S., Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), 495–550.CrossRefGoogle Scholar
[Tia97]Tian, G., Kahler–Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), 1–37.CrossRefGoogle Scholar
[Wit09]Witt Nyström, D., Transforming metrics on a line bundle to the Okounkov body, Preprint (2009), math.CV/0903.5167.Google Scholar