On Zhang's semipositive metrics

TL;DR

This paper extends the concept of semipositive metrics from proper varieties to more general non-archimedean analytic spaces, establishing key properties and a closure result for semipositive model metrics.

Contribution

It generalizes Zhang's semipositive metrics to non-archimedean analytic spaces and proves their density and closure properties in this broader setting.

Findings

Piecewise $ extbf{Q}$-linear metrics are dense among continuous metrics.

Algebraic, formal, and piecewise linear metrics coincide on proper schemes.

Semipositive model metrics form a closed set under pointwise convergence.

Abstract

Zhang introduced semipositive metrics on a line bundle of a proper variety. In this paper, we generalize such metrics for a line bundle $L$ of a paracompact strictly $K$-analytic space $X$ over any non-archimedean field $K$. We prove various properties in this setting such as density of piecewise $\mathbb{Q}$-linear metrics in the space of continuous metrics on $L$. If $X$ is proper scheme, then we show that algebraic, formal and piecewise linear metrics are the same. Our main result is that on a proper scheme $X$ over an arbitrary non-archimedean field $K$, the set of semipositive model metrics is closed with respect to pointwise convergence generalizing a result from Boucksom, Favre and Jonsson where $K$ was assumed to be discretely valued with residue characteristic $0$.