Dolbeault Cohomology of Graphs and Berkovich Curves

TL;DR

This paper develops Dolbeault cohomology theory for graphs and Berkovich curves, establishing duality results and computing cohomology groups for non-Archimedean analytic curves.

Contribution

It introduces real-valued forms on graphs, computes their Dolbeault cohomology, and extends these results to Berkovich curves using skeletons and duality principles.

Findings

Computed Dolbeault cohomology for weighted metric graphs.

Proved Poincaré duality for these cohomology groups.

Extended results to non-Archimedean curves, confirming duality in proper cases.

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 Poincar\'e 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 Poincar\'e duality when $X$ is proper.