A Poincar\'e 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 diff\'erentielles r\'eelles et courants sur les espaces de Berkovich' using superforms on polyhedral complexes. We prove a Poincar\'e lemma for these superforms and use it to also prove a Poincar\'e 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.