# Tropical Hodge numbers of non-archimedean curves

## TL;DR

This paper investigates the tropical Dolbeault cohomology of non-archimedean curves, establishing conditions for Poincaré duality, and computes tropical Hodge numbers, revealing infinite dimensions in some cases.

## Contribution

It provides a precise criterion for Poincaré duality in tropical Dolbeault cohomology of non-archimedean curves and introduces an exponential sequence relating cohomologies.

## Key findings

- Poincaré duality holds when the residue field is an algebraic closure of a finite field.
- Tropical (1,1)-Dolbeault cohomology can be infinite dimensional over complex residue fields.
- Dimensions of tropical Hodge numbers are computed for open subsets of 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 Poincar\'e 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 where the residue field is the field of complex numbers, 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 Poincar\'e duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.