Tropical Skeletons

TL;DR

This paper explores the relationship between tropical and analytic geometry for subschemes of toric varieties over non-Archimedean fields, introducing the tropical skeleton and analyzing its properties and continuity aspects.

Contribution

It defines the tropical skeleton in the Berkovich space, develops criteria for its closure and limit points, and relates it to known structures like semistable pairs and the Helm--Katz parameterizing complex.

Findings

Tropical skeleton is characterized by Shilov boundary points.

Criteria for the closure and continuity of the tropicalization map are established.

Tropical skeleton coincides with the skeleton of a semistable pair in certain cases.

Abstract

In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the tropical skeleton of $X$ as the subset of the associated Berkovich space $X^{\rm an}$ which collects all Shilov boundary points in the fibers of the Kajiwara--Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When $X$ is sch\"on and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm--Katz.