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Vanishing sequences and Okounkov bodies

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Abstract

We define and study the vanishing sequence along a real valuation of sections of a line bundle on a normal projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure; in particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.

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Acknowledgments

We are grateful to Bo Berndtsson, Lawrence Ein, Patrick Graf, Daniel Greb, and Rob Lazarsfeld for helpful discussions. During this project, Sébastien Boucksom was partially supported by the ANR projects MACK and POSITIVE. Alex Küronya was supported in part by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds”, and the OTKA Grants 77476 and 81203 by the Hungarian Academy of Sciences. Tomasz Szemberg’s research was partly supported by NCN grant UMO-2011/01/B/ ST1/04875. Catriona Maclean was supported by the ANR project CLASS. Part of this work was done while the second author was visiting the Uniwersytet Pedagogiczny in Cracow, and while the second and third authors were visiting the Université Pierre et Marie Curie in Paris. We would like to use this opportunity to thank Anreas Höring for the invitation to UPMC and both institutions for the excellent working conditions.

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Author notes

  1. Alex Küronya

    Present address: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstraße 1, 79104, Freiburg, Germany

Authors and Affiliations

  1. Institut de Mathématiques, CNRS–Université Paris 6, 75251, Paris Cedex 05, France

    Sébastien Boucksom

  2. Department of Algebra, Mathematical Institute, Budapest University of Technology and Economics, Pf. 91, 1521, Budapest, Hungary

    Alex Küronya

  3. Institut Fourier, CNRS UMR 5582 Université de Grenoble, 100 rue des Maths, 38402, Saint-Martin d’Héres cedex, France

    Catriona Maclean

  4. Instytut Matematyki UP, Podchorążych 2, 30-084, Kraków, Poland

    Tomasz Szemberg

Authors
  1. Sébastien Boucksom
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  2. Alex Küronya
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  3. Catriona Maclean
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  4. Tomasz Szemberg
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Corresponding author

Correspondence to Alex Küronya.

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Boucksom, S., Küronya, A., Maclean, C. et al. Vanishing sequences and Okounkov bodies. Math. Ann. 361, 811–834 (2015). https://doi.org/10.1007/s00208-014-1081-z

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