Analytification and Tropicalization over non-archimedean fields

TL;DR

This paper reviews recent advances in the relationship between tropical geometry and non-archimedean Berkovich spaces, highlighting key results, constructions of skeleta, and comparisons between tropicalizations and analytic structures.

Contribution

It synthesizes recent developments in higher-dimensional theory, including constructions of generalized skeleta and comparison results between Berkovich spaces and tropicalizations.

Findings

Comparison of tropical Grassmannian and analytic Grassmannian for planes.

Construction of generalized skeleta in Berkovich spaces.

Results on the slope formula and polyhedral substructures.

Abstract

This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of curves, we explain a result by Cueto, H\"abich and the author comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of the general higher-dimensional theory developed by Gubler, Rabinoff and the author. In particular, we explain the construction of generalized skeleta in which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the comparison between polyhedral substructures of Berkovich spaces and tropicalizations.