Hahn analytification and connectivity of higher rank tropical varieties

TL;DR

This paper proves that the tropicalization of connected varieties over higher rank valued fields is path connected, extending known results and establishing new links between Hahn analytifications and other non-Archimedean analytic spaces.

Contribution

It introduces Hahn analytifications for higher rank valued fields and relates them to higher rank tropicalization via an inverse limit theorem, extending classical non-Archimedean results.

Findings

Tropicalization of connected higher rank varieties is path connected.

Hahn analytification relates to higher rank tropicalization through inverse limits.

Comparison established between Hahn, Huber, Berkovich analytifications and stable completion.

Abstract

We show that the tropicalization of a connected variety over a higher rank valued field is a path connected topological space. This establishes an affirmative answer to a question posed by Banerjee. Higher rank tropical varieties are studied as the images of "Hahn analytifications", introduced in this paper. A Hahn analytification is a space of valuations on a scheme over a higher rank valued field. We prove that the Hahn analytification is related to higher rank tropicalization by means of an inverse limit theorem, extending well-known results in the non-Archimedean case. We also establish comparison results between the Hahn analytification and the Huber and Berkovich analytifications, as well as the Hrushovski-Loeser stable completion.