Tropicalization is a non-Archimedean analytic stack quotient

TL;DR

This paper establishes that the Kajiwara-Payne tropicalization map for non-Archimedean analytic spaces can be understood as a stack quotient, extending classical results from complex toric varieties to the non-Archimedean setting.

Contribution

It introduces a geometric framework for non-Archimedean analytic stacks, including groupoids and quotients, and proves the tropicalization map as a stack quotient.

Findings

Tropicalization map is a non-Archimedean analytic stack quotient.

Develops foundations for non-Archimedean analytic stacks.

Provides a geometric perspective on tropicalization in non-Archimedean geometry.

Abstract

For a complex toric variety $X$ the logarithmic absolute value induces a natural retraction of $X$ onto the set of its non-negative points and this retraction can be identified with a quotient of $X(\mathbb{C})$ by its big real torus. We prove an analogous result in the non-Archimedean world: The Kajiwara-Payne tropicalization map is a non-Archimedean analytic stack quotient of $X^{an}$ by its big affinoid torus. Along the way, we provide foundations for a geometric theory of non-Archimedean analytic stacks, particularly focussing on analytic groupoids and their quotients, the process of analytification, and the underlying topological spaces of analytic stacks.