Cohomology of real-valued differential forms on Berkovich analytic spaces

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In algebraic geometry one studies the geometry of the set of solutions of a family of polynomial equations. One method to study integral solutions of such systems of equations is Arakelov theory. It was Arakelov's great insight that to study these solutions, it is very helpful to combine algebraic geometry at the prime numbers, often called finite places, with analytic geometry over the complex numbers. It has always been the hope in Arakelov theory that one can use analytic geometry also at finite places. In particular, one needs a notion of real-valued differential forms at such a finite place. In the 1990s, Berkovich introduced suitable analytic spaces, called Berkovich analytic spaces. In 2012 Chambert-Loir and Ducros introduced smooth real-valued differential forms on Berkovich analytic spaces. Chambert-Loir and Ducros, Gubler and Künnemann as well as Liu showed first results in applying these differential forms in Arakelov theory. My own previous results include a Poincaré lemma for these differential forms, which was crucially used in Liu’s work. Further, in joint work with V. Wanner, we showed that the cohomology with respect to smooth real-valued differential forms of Mumford curves satisfies Poincaré duality and used this to completely calculate that cohomology for Mumford curves. The goal of my research project is to study these smooth real-valued differential forms in a general context and prove results about their cohomology, which are analogous to the results over the complex numbers. In particular, I want to prove that the cohomology of curves satisfies Poincaré duality. Poincaré duality is one of the basic properties of smooth differential forms over the complex numbers. It is both useful in theoretical applications as well as in concrete calculations of the cohomology. Since the definition of smooth-real valued differential forms uses tropical geometry and previous work shows direct relations to invariants in tropical geometry, studying questions in tropical geometry will also be part of the project. In said previous work, which was joint work with K. Shaw and J. Smacka, we further showed that smooth tropical varieties satisfy Poincaré duality. I want to show that more tropical spaces than currently known satisfy Poincaré duality. Also I want to prove that certain tropical spaces, and in particular smooth projective tropical varieties, satisfy symmetry in Hodge numbers.