Poincar\'e duality for tropical Dolbeault cohomology of non-archimedean   Mumford curves
  We calculate the tropical Dolbeault cohomology for the analytifications of the projective line and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincar\'e duality and behaves analogously to the cohomology of curves over the complex numbers. Further, we give a complete calculation of the dimension of the cohomology on a basis of the topology. 
Philipp Jell and Veronika Wanner
math.AG