A tropical approach to nonarchimedean Arakelov geometry

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 $\delta$-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 $\delta$-form. We introduce the associated Monge–Ampère measure $\mu$ as a wedge-power of this first Chern $\delta$-form and we show that $\mu$ is equal to the corresponding Chambert-Loir measure. The $\ast$-product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of $\delta$-forms, we obtain a nonarchimedean analogue at least in the case of divisors. We use it to compute nonarchimedean local heights of proper varieties.