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arXiv · 1410.2216

Tropicalization is a non-Archimedean analytic stack quotient

Martin Ulirsch
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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.

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Martin Ulirsch. 2014-10-08. Tropicalization is a non-Archimedean analytic stack quotient. https://arxiv.org/abs/1410.2216

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