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.

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The author’s research was supported in part by funds from BSF grant 201025 and NSF grants DMS0901278 and DMS1162367.

Received 24 July 2015

Published 9 November 2017