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

TL;DR

This paper explores the metric and geometric properties of non-Archimedean norms on section rings of polarized varieties, introducing a new action of filtrations that generalizes previous work and reveals non-Archimedean geodesic structures.

Contribution

It establishes a jointly d_1-contracting action of filtrations on graded norms, leading to the construction of non-Archimedean geodesic rays and flats, extending prior results.

Findings

Existence of a d_1-contracting action of filtrations on graded norms

Construction of non-Archimedean geodesic rays and flats

Convergence of limit measures along geodesic rays

Abstract

This article is concerned with the metric study of a construction of G\'erardin 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 Nystr\"om. 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.