Non-Archimedean volumes of metrized nef line bundles
| Autor: | Boucksom, Sébastien, Gubler, Walter, Martin, Florent |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | Ãpijournal de Géométrie Algébrique, Volume 5 (October 5, 2021) epiga:6908 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.46298/epiga.2021.6908 |
| Popis: | 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 $K$. Comment: Published version, 34 pages |
| Databáze: | arXiv |
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