Forms on Berkovich spaces based on harmonic tropicalizations

TL;DR

This paper develops a new framework for harmonic tropicalizations on Berkovich spaces, introducing tropical skeletons and expanding differential forms with improved cohomological properties.

Contribution

It presents a novel approach to harmonic tropicalizations, enlarging the sheaf of differential forms with better cohomological behavior on Berkovich spaces.

Findings

Tropical skeletons are introduced for Berkovich spaces.

Harmonic tropicalizations produce balanced tropical varieties.

Enhanced sheaves of differential forms with improved cohomological properties.

Abstract

We introduce tropical skeletons for Berkovich spaces based on results of Ducros. Then we study harmonic functions on good strictly analytic spaces over a non-trivially valued non-Archimedean field. Chambert-Loir and Ducros introduced bigraded sheaves of smooth real-valued differential forms on Berkovich spaces by pulling back Lagerberg forms with respect to tropicalization maps. We give a new approach in which we allow pullback by more general harmonic tropicalizations to get a larger sheaf of differential forms with essentially the same properties, but with a better cohomological behavior. A crucial ingredient is that tropical varieties arising from harmonic tropicalization maps are balanced.