On Zhang's Semipositive Metrics

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 .