Non-Archimedean volumes of metrized nef line bundles
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arXiv
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| Format: | Preprint |
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2020
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| author | Boucksom, Sébastien Gubler, Walter Martin, Florent |
| author_facet | Boucksom, Sébastien Gubler, Walter Martin, Florent |
| contents | 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 $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_06986 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Non-Archimedean volumes of metrized nef line bundles Boucksom, Sébastien Gubler, Walter Martin, Florent Algebraic Geometry Number Theory Primary 32P05, Secondary 14G22, 32U15, 32W20 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 $K$. |
| title | Non-Archimedean volumes of metrized nef line bundles |
| topic | Algebraic Geometry Number Theory Primary 32P05, Secondary 14G22, 32U15, 32W20 |
| url | https://arxiv.org/abs/2011.06986 |