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 diff,rentielles r,elles et courants sur les espaces de Berkovich. arXiv:1204.6277, 2012) using superforms on polyhedral complexes. We prove a Poincar, lemma for these superforms and use it to also prove a Poincar, 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