On Zhang's semipositive metrics

Walter Gubler; Florent Martin

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arXiv · 1608.08030

On Zhang's semipositive metrics

Walter Gubler·Florent Martin

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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 $0$.

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Walter Gubler, Florent Martin. 2016-08-29. On Zhang's semipositive metrics. https://arxiv.org/abs/1608.08030

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