Limits of tropicalizations
type: Item
instance of: scholarly article
title: Limits of tropicalizations
zbMATH Open document ID: 1319.14059
author: Tyler Foster
author: Philipp Gross
author: Sam Payne
published in: Israel Journal of Mathematics
publication date: 2014-10-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.2718
review text: This paper is a follow-up to the third author's now quite well-known paper ``Analytification is the limit of all tropicalizations'', [\textit{S. Payne}, Math. Res. Lett. 16, No. 2--3, 543--556 (2009; Zbl 1193.14077)]. In the latter, Payne establishes a fundamental link between tropicalization and non-archimedean analytification (in the sense of Berkovich) which has served as a focal point in many applications of tropical geometry and subsequent developments in the subject. It is difficult to overstate the importance of this paper of Payne, even though he seems to imply in it that all the results were known in some form in much earlier literature. Regardless of that point, this paper cleaned up the story, elucidated the connection, and established a fundamentally important bridge between two rapidly developing fields. The main result from that paper is the following. The tropicalization of a variety depends on the choice of an embedding in a toric variety defined over a valued field. Since tropicalization is functorial with respect to toric morphisms, as one varies the embedding (while keeping the valued field fixed), one obtains an inverse system of tropicalizations, and the limit of this system (when one topologizes the tropicalizations appropriately using the Euclidean topology) is homeomorphic to the Berkovich analytification of the original variety if the latter is quasi-projective. If the variety is affine then one can use a much smaller system: instead of embeddings in arbitrary toric varieties, it suffices to consider embeddings in affine space. The present paper looks closer at this result and derives more precise conditions on the inverse system in order to guarantee that the limit remains the Berkovich analytification. See the introduction for the main statement. The idea is, in essence, to see exactly what properties were really used in Payne's argument and to adapt a more general argument based on an abstraction of these properties. This leads to two imp…
reviewed by: Noah Giansiracusa
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 6361319
zbMATH Keywords: tropicalization
zbMATH Keywords: analytification
zbMATH Keywords: Berkovich
zbMATH Keywords: inverse limit
zbMATH Keywords: toric
zbMATH Keywords: non-archimedean
MaRDI profile type: Publication
OpenAlex ID: W2963912243
arXiv ID: 1211.2718
cites work: Q3998951
cites work: Smooth \(p\)-adic analytic spaces are locally contractible
cites work: Q3155110
cites work: Every morphism is the restriction of a toric one.
cites work: Nonarchimedean geometry, tropicalization, and metrics on curves
cites work: The functor of a smooth toric variety
cites work: The Picard group of a compact toric variety
cites work: Introduction to Toric Varieties. (AM-131)
cites work: ON WłODARCZYK'S EMBEDDING THEOREM
cites work: Equivariant Embeddings into Smooth Toric Varieties
cites work: Non-Archimedean tame topology and stably dominated types
cites work: On embeddings into toric prevarieties
cites work: Q3525465
cites work: Q5482626
cites work: Analytification is the limit of all tropicalizations
cites work: Toric vector bundles, branched covers of fans, and the resolution property
cites work: Tropical analytic geometry, Newton polygons, and tropical intersections
cites work: Toroidal geometry and non-archimedean analytic geometry. Application to the homotopy type of certain formal schemes
cites work: Q4277786
DOI: 10.1007/S11856-014-1051-X
description: scientific article; zbMATH DE number 6361319