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Okounkov bodies of filtered linear series

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  04 May 2011

Sébastien Boucksom  and
Huayi Chen
Sébastien Boucksom
Affiliation:
IMJ Université Pierre et Marie Curie, Case 247, 4 place Jussieu, 75252 Paris Cedex, France (email: boucksom@math.jussieu.fr)
Huayi Chen
Affiliation:
Institut de Mathématiques de Jussieu, IMJ Université Paris 7, 175 rue du Chevaleret, 75013 Paris, France (email: chenhuayi@math.jussieu.fr)
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Abstract

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We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.

Keywords

big line bundleOkounkov bodyarithmetic volumesuccessive minima

MSC classification

Secondary: 14G25: Global ground fields 11G50: Heights
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 4 , July 2011 , pp. 1205 - 1229
Copyright
Copyright © Foundation Compositio Mathematica 2011

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