A Poincaré lemma for real-valued differential forms on Berkovich spaces

Real-valued differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros in 'Formes différentielles réelles et courants sur les espaces de Berkovich' using superforms on polyhedral complexes. We prove a Poincaré lemma for these superforms and use it to also prove a Poincaré lemma for real-valued differential forms on Berkovich spaces. For superforms we further show finite dimensionality for the associated de Rham cohomology on polyhedral complexes in all (bi-)degrees. We also show finite dimensionality for the real-valued de Rham cohomology of the analytification of an algebraic variety in some bidegrees.

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