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arXiv · 2111.05747

Dolbeault Cohomology of Graphs and Berkovich Curves

Walter Gubler·Philipp Jell·Joseph Rabinoff
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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.

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Walter Gubler, Philipp Jell, Joseph Rabinoff. 2021-11-10. Dolbeault Cohomology of Graphs and Berkovich Curves. https://arxiv.org/abs/2111.05747

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