Tropical Hodge numbers of non-archimedean curves

Abstract

We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincare duality. The condition is always satisfied when the residue field of the non-archimedean base field is the algebraic closure of a finite field. We also show that for curves over fields with residue field ℂ, the tropical (1, 1)-Dolbeault cohomology can be infinite dimensional. Our main new ingredient is an exponential type sequence that relates tropical Dolbeault cohomology to the cohomology of the sheaf of harmonic functions. As an application of our Poincare duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.