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
-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
-form.
We introduce the associated Monge–Ampère measure
as a wedge-power of
this first Chern
-form
and we show that
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
-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