Poincar\'e duality for tropical Dolbeault cohomology of non-archimedean Mumford curves

TL;DR

This paper computes the tropical Dolbeault cohomology for non-archimedean Mumford curves, demonstrating Poincaré duality and providing explicit dimension calculations, thus bridging non-archimedean and complex algebraic geometry.

Contribution

It establishes Poincaré duality for tropical Dolbeault cohomology of Mumford curves and provides explicit dimension formulas, advancing understanding of non-archimedean analytic geometry.

Findings

Cohomology satisfies Poincaré duality.

Explicit dimension formulas for cohomology groups.

Analogy with complex algebraic curve cohomology.

Abstract

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.