Filtrations and asymptotic geometry of non-Archimedean norms on section rings
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914363642740736 |
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| author | Reboulet, Rémi |
| author_facet | Reboulet, Rémi |
| contents | This article is concerned with the metric study of a construction of Gérardin of the action of the boundary at infinity of the space of norms on a non-Archimedean vector space, and its generalisation to graded algebras. Namely, given (X,L) a polarised variety over an arbitrary non-Archimedean field, we show that there is a jointly d_1-contracting action of the space of filtrations of the section ring R(X,L) on the space of graded norms on R(X,L). This naturally yields non-Archimedean geodesic rays and infinite-dimensional flats in this setting, generalising previous work of the author and Witt Nyström. It is further shown that relative limit measures converge along geodesic rays, providing a result on the d_p-radial geometry of graded norms, analogous to a recent result of Finski in the Archimedean case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Filtrations and asymptotic geometry of non-Archimedean norms on section rings Reboulet, Rémi Algebraic Geometry Metric Geometry 32P05, 32P99, 32U15, 14G22 This article is concerned with the metric study of a construction of Gérardin of the action of the boundary at infinity of the space of norms on a non-Archimedean vector space, and its generalisation to graded algebras. Namely, given (X,L) a polarised variety over an arbitrary non-Archimedean field, we show that there is a jointly d_1-contracting action of the space of filtrations of the section ring R(X,L) on the space of graded norms on R(X,L). This naturally yields non-Archimedean geodesic rays and infinite-dimensional flats in this setting, generalising previous work of the author and Witt Nyström. It is further shown that relative limit measures converge along geodesic rays, providing a result on the d_p-radial geometry of graded norms, analogous to a recent result of Finski in the Archimedean case. |
| title | Filtrations and asymptotic geometry of non-Archimedean norms on section rings |
| topic | Algebraic Geometry Metric Geometry 32P05, 32P99, 32U15, 14G22 |
| url | https://arxiv.org/abs/2508.16322 |