Positivity properties of metrics and delta-forms

TL;DR

This paper explores positivity properties of delta-forms and delta-currents on Berkovich spaces, linking them to plurisubharmonicity and semipositivity, with applications to various types of metrics on algebraic varieties.

Contribution

It introduces new positivity notions for delta-forms, establishes their equivalence to Zhang's semipositivity in certain cases, and provides convex geometric criteria for plurisubharmonicity.

Findings

Positivity notions for delta-forms are introduced and studied.

Many positivity notions are equivalent to Zhang's semipositivity for formal metrics.

Plurisubharmonicity can be tested via convex geometry on tropical charts.

Abstract

In previous work, we have introduced delta-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern delta-forms. In this paper, we investigate positivity properties of delta-forms and delta-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang's semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.