Metrized ample line bundles in non-Archimedean geometry II
Yanbo Fang
TL;DR
The paper develops a non-Archimedean analytic framework to study semipositive metrics on ample line bundles via the normed section algebra, introducing Shilov finite metrics and the equidistribution measure. It proves an arithmetic Hilbert–Samuel formula at a local place by analyzing determinant-norm distortions in normed exact sequences, first in the Shilov-finite case and then by approximation to general semipositive metrics. A core result is that the asymptotic distortion equals the integral of $\log|s|$ against the equidistribution measure, which matches the Monge–Ampère measure for continuous metrics, linking analytic and algebraic data. The paper also connects these local non-Archimedean analyses to global HS formulas and discusses the continuity, differentiation, and potential extensions of the equidistribution/Monge–Ampère framework, with concrete examples in locally affinoid settings and envelope-constructed metrics.
Abstract
We introduce a class of semipositive metrics on ample line bundles in non-Archimedean geometry, called Shilov finite metrics. We calculate the determinant metric distorsion in the exact sequence induced by a global section using non-Archimedean norm reduction techniques. This leads to an analytic proof to the arithmetic Hilbert-Samuel formula over a local place for a semipositively metrized ample line bundle. We define the equidistribution measure associated to a Shilov finite metric and solve the corresponding inverse problem.