Sébastien Boucksom ; Walter Gubler ; Florent Martin - Non-Archimedean volumes of metrized nef line bundles

epiga:6908 - Épijournal de Géométrie Algébrique, October 5, 2021, Volume 5 - https://doi.org/10.46298/epiga.2021.6908
Non-Archimedean volumes of metrized nef line bundles

Authors: Sébastien Boucksom ; Walter Gubler ; Florent Martin

Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a non-trivially valued non-Archimedean field $K$. Roughly speaking, the non-Archimedean volume of a continuous metric on the Berkovich analytification of $L$ measures the asymptotic growth of the space of small sections of tensor powers of $L$. For a continuous semipositive metric on $L$ in the sense of Zhang, we show first that the non-Archimedean volume agrees with the energy. The existence of such a semipositive metric yields that $L$ is nef. A second result is that the non-Archimedean volume is differentiable at any semipositive continuous metric. These results are known when $L$ is ample, and the purpose of this paper is to generalize them to the nef case. The method is based on a detailed study of the content and the volume of a finitely presented torsion module over the (possibly non-noetherian) valuation ring of $K$.

Volume: Volume 5
Published on: October 5, 2021
Accepted on: October 5, 2021
Submitted on: November 16, 2020
Keywords: Mathematics - Algebraic Geometry,Mathematics - Number Theory,Primary 32P05, Secondary 14G22, 32U15, 32W20


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