A Tropical Characterization of Algebraic Subvarieties of Toric Varieties over Non-Archimedean Fields

J. Math. Sci. Univ. Tokyo
Vol. 25 (2018), No. 2, Page 129-147.

Mikami, Ryota
A Tropical Characterization of Algebraic Subvarieties of Toric Varieties over Non-Archimedean Fields
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Abstract:
We study the tropicalizations of analytic subvarieties of normal toric varieties over complete non-archimedean valuation fields. We show that a Zariski closed analytic subvariety of a normal toric variety is algebraic if its tropicalization is a finite union of polyhedra. Previously, the converse direction was known by the theorem of Bieri and Groves. Over the field of complex numbers, Madani, L. Nisse, and M. Nisse proved similar results for analytic subvarieties of tori.

Keywords: Tropical geometry, rigid analytic geometry

Mathematics Subject Classification (2010): Primary 14T05; Secondary 14G22; Tertiary 14M25
Mathematical Reviews Number: MR3792788

Received: 2017-08-02