A tropical approach to nonarchimedean Arakelov geometry

Gubler, Walter; Künnemann, Klaus

doi:10.2140/ant.2017.11.77

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A tropical approach to nonarchimedean Arakelov geometry

Walter Gubler and Klaus Künnemann

Vol. 11 (2017), No. 1, 77–180

DOI: 10.2140/ant.2017.11.77

Abstract

Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called $δ$ -forms on the nonarchimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincaré–Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a $δ$ -form. We introduce the associated Monge–Ampère measure $μ$ as a wedge-power of this first Chern $δ$ -form and we show that $μ$ is equal to the corresponding Chambert-Loir measure. The $*$ -product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of $δ$ -forms, we obtain a nonarchimedean analogue at least in the case of divisors. We use it to compute nonarchimedean local heights of proper varieties.

Keywords

differential forms on Berkovich spaces, Chambert-Loir measures, tropical intersection theory, nonarchimedean Arakelov theory

Mathematical Subject Classification 2010

Primary: 14G40

Secondary: 14G22, 14T05, 32P05

Milestones

Received: 19 October 2015

Revised: 13 September 2016

Accepted: 13 November 2016

Published: 23 January 2017

Authors

Walter Gubler
Fakultät für Mathematik Universität Regensburg Universitätsstraße 31 D-93040 Regensburg Germany

Klaus Künnemann
Fakultät für Mathematik Universität Regensburg Universitätsstraße 31 D-93040 Regensburg Germany

Vol. 11, No. 1, 2017