Dolbeault Cohomology of Graphs and Berkovich Curves

Abstract

We introduce real-valued $(p,q)$-forms on weighted metric graphs with boundary similar to Lagerberg forms on polyhedral spaces. We compute the Dolbeault cohomology and prove Poincaré duality. Using Thuillier's thesis, the skeleton of a strictly semistable formal curve is canonically a weighted metric graph with boundary. We use that and our companion paper on weakly smooth forms to compute the Dolbeault cohomology for weakly smooth forms on any non-Archimedean compact rig-smooth analytic curve $X$, and prove Poincaré duality when $X$ is proper.

56 pages, 1 figure. Comments welcome!

References in corpus (1)

• Poincaré duality for tropical Dolbeault cohomology of non-archimedean Mumford curves