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Filtrations and asymptotic geometry of non-Archimedean norms on section rings

Rémi Reboulet

Abstract

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.

Filtrations and asymptotic geometry of non-Archimedean norms on section rings

Abstract

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.

Paper Structure

This paper contains 25 sections, 51 theorems, 162 equations.

Table of Contents

  1. The Goldman--Iwahori space and the Gérardin action.
  2. The graded setting.
  3. Strategy of proof.
  4. Thoughts.
  5. Organisation of the paper.
  6. Acknowledgements.
  7. Asymptotic geometry of the space of norms.
  8. Preliminaries: the Goldman--Iwahori space.
  9. Distances, volumes, determinants.
  10. The action of $\mathcal{N}_0(V)$.
  11. Continuity of the action in $d_\infty$.
  12. Continuity of the action in $d_1$.
  13. Geodesic segments.
  14. Geodesic rays.
  15. Asymptotic geometry of the space of graded norms.
  16. ...and 10 more sections

Key Result

Lemma 1.1

Let $S\subset \mathcal{N}(V)$ be a subset, and assume that there exists a finite-valued function $f:V\to\mathbb{R}$ such that, for all $\left\|\cdot\right\|\in S$ and all $v\in V$, $\left\|v\right\|\leq f(v)$. Then is a norm in $\mathcal{N}(V)$.

Theorems & Definitions (100)

  • Lemma 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Proposition 1.6: boueri
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • Definition 1.9
  • ...and 90 more