Limits of tropicalizations

TL;DR

This paper establishes criteria for when the limit of tropicalizations of a scheme over a nonarchimedean field matches its analytification, and applies this to subschemes of toric varieties, extending previous results.

Contribution

It provides general conditions for tropicalization limits to be homeomorphic to analytifications and extends known results to arbitrary closed subschemes of toric varieties.

Findings

Tropicalization limits can be homeomorphic to analytifications under certain criteria.

The analytification of any closed subscheme of a toric variety equals the limit of its tropicalizations.

Generalization of previous results from quasiprojective to arbitrary closed subschemes.

Abstract

We give general criteria under which the limit of a system of tropicalizations of a scheme over a nonarchimedean field is homeomorphic to the analytification of the scheme. As an application, we show that the analytification of an arbitrary closed subscheme of a toric variety is naturally homeomorphic to the limit of its tropicalizations, generalizing an earlier result of the third author for quasiprojective varieties.