Algebra & Number Theory

A tropical approach to nonarchimedean Arakelov geometry

Walter Gubler and Klaus Künnemann

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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.

Article information

Algebra Number Theory, Volume 11, Number 1 (2017), 77-180.

Received: 19 October 2015
Revised: 13 September 2016
Accepted: 13 November 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30]
Secondary: 14G22: Rigid analytic geometry 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20] 32P05: Non-Archimedean analysis (should also be assigned at least one other classification number from Section 32 describing the type of problem)

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


Gubler, Walter; Künnemann, Klaus. A tropical approach to nonarchimedean Arakelov geometry. Algebra Number Theory 11 (2017), no. 1, 77--180. doi:10.2140/ant.2017.11.77.

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