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Distribution of logarithmic spectra of the equilibrium energy

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Abstract

Let L be a big invertible sheaf on a projective variety defined on a complete valued field (such as the field \({\mathbb{C}}\) of complex numbers or a complete non-archimedean field), equipped with two continuous metrics. By using the ideas in Arakelov geometry, we prove that the distribution of the eigenvalues of the transition matrix between the L 2 norms on H 0(X,nL) with respect to the two metriques converges (in law) as n goes to infinity to a Borel probability measure on \({\mathbb{R}}\). This result can be thought of as a generalization of the existence of the energy at the equilibrium as a limit, or an extension of Berndtsson’s results to the more general context of graded linear series and a more general class of line bundles.

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  1. Institut Fourier, Université Grenoble Alpes, 38000, Grenoble, France

    Huayi Chen & Catriona Maclean

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  1. Huayi Chen
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  2. Catriona Maclean
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Correspondence to Huayi Chen.

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Chen, H., Maclean, C. Distribution of logarithmic spectra of the equilibrium energy. manuscripta math. 146, 365–394 (2015). https://doi.org/10.1007/s00229-014-0712-8

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  • DOI: https://doi.org/10.1007/s00229-014-0712-8

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