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Mathematical Research Letters
Volume 24 (2017)
Number 4
Tropicalization is a non-Archimedean analytic stack quotient
Pages: 1205 – 1237
DOI: https://dx.doi.org/10.4310/MRL.2017.v24.n4.a12
Author
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, focussing on analytic groupoids and their quotients, the process of analytification, and the underlying topological spaces of analytic stacks.
The author’s research was supported in part by funds from BSF grant 201025 and NSF grants DMS0901278 and DMS1162367.
Received 24 July 2015
Accepted 18 July 2016
Published 9 November 2017