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

2025/08/22 by Reboulet, Rémi
#14G22 #32P05 #32P99 #32U15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.16322

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 d₁-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.

Citations

• Lines in the space of Kähler metrics
• Infinite-dimensional flats in the space of positive metrics on an ample line bundle
• Measures of finite energy in pluripotential theory: a synthetic approach
• Submultiplicative norms and filtrations on section rings
• On the metric structure of section ring
• A non-Archimedean approach to K-stability, II: divisorial stability and openness
• A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations
• The space of finite-energy metrics over a degeneration of complex manifolds
• Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy
• The asymptotic Fubini-Study operator over general non-Archimedean fields
• K-stability of Fano varieties via admissible flags
• Variational and non-Archimedean aspects of the Yau--Tian--Donaldson\n conjecture
• Spaces of norms, determinant of cohomology and Fekete points in\n non-Archimedean geometry
• Global pluripotential theory over a trivially valued field
• A variational approach to the Yau-Tian-Donaldson conjecture
• Distribution of logarithmic spectra of the equilibrium energy
• Singular semipositive metrics in non-Archimedean geometry
• Filtrations and test-configurations
• Test Configurations for K-Stability and Geodesic Rays
• L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld : demonstration du resultat principal
• Szego kernels and a theorem of Tian

Related

• The asymptotic Fubini-Study operator over general non-Archimedean fields
• Submultiplicative norms and filtrations on section rings
• Non-Archimedean metric extension for semipositive line bundles
• A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations
• On the metric structure of section ring