Tropical cohomology with integral coefficients for analytic spaces

TL;DR

This paper develops a framework for tropical Dolbeault cohomology with integral coefficients on Berkovich analytic spaces, enabling class pullback and non-triviality checks, using extended tropicalization of toric varieties.

Contribution

It introduces tropical cohomology with integral coefficients and provides methods to relate it to tropical Dolbeault cohomology on Berkovich spaces.

Findings

Construction for pulling back classes in tropical cohomology

Definition of tropical cohomology with integral coefficients

Computations of tropical cohomology with integral coefficients

Abstract

We study tropical Dolbeault cohomology for Berkovich analytic spaces, as defined by Chambert-Loir and Ducros. We provide a construction that lets us pull back classes in tropical cohomology to classes in tropical Dolbeault cohomology as well as check whether those classes are non-trivial. We further define tropical cohomology with integral coefficients on the Berkovich space and provide some computations. Our main tool is extended tropicalization of toric varieties as introduced by Kajiwara and Payne.