Gubler, Walter; Martin, Florent

On Zhang's Semipositive Metrics

Doc. Math. 24, 331-372 (2019)
DOI: 10.25537/dm.2019v24.331-372


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 \(0\).

Mathematics Subject Classification

14G40, 14G22


Arakelov geometry, non-Archimedean geometry, model metrics, plurisubharmonic model functions, divisorial points


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Gubler, Walter
Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 31, D-93040 Regensburg, Germany
Martin, Florent
Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 31, D-93040 Regensburg, Germany