Mathematical Research Letters

Volume 24 (2017)

Number 4

Tropicalization is a non-Archimedean analytic stack quotient

Pages: 1205 – 1237



Martin Ulirsch (Department of Mathematics, Brown University, Providence, Rhode Island, U.S.A.; and Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada)


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