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manuscripta mathematica

, Volume 151, Issue 3–4, pp 353–374 | Cite as

Hahn analytification and connectivity of higher rank tropical varieties

  • Tyler Foster
  • Dhruv RanganathanEmail author
Article

Abstract

We show that the tropicalization of a connected variety over a higher rank valued field is a path connected topological space. This establishes an affirmative answer to a question posed by Banerjee (J Reine Angew Math (Crelle’s J) 2015:71–87, 2013). Higher rank tropical varieties are studied as the images of “Hahn analytifications”, introduced in this paper. A Hahn analytification is a space of valuations on a scheme over a higher rank valued field. We prove that the Hahn analytification is related to higher rank tropicalization by means of an inverse limit theorem, extending well-known results in the non-Archimedean case. We also establish comparison results between the Hahn analytification and the Huber and Berkovich analytifications, as well as the Hrushovski–Loeser stable completion.

Mathematics Subject Classification

14T05 32P05 

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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