Analytification is the limit of all tropicalizations

By Sam Payne. 2008

216 citations recorded by the source corpus.

We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/2971eda199acabfde2a50d31a737eae1b9643fa4

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In Computational Theory and Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “Section 7, finally, is concerned with a comparison of our construction with the well-known embedded tropicalization for (subvarieties of) toric varieties in the sense of Kajiwara and Payne (see [29] and [41]) and a proof of Corollary 1.”
Algebraic Geometry Over Hyperrings (2015, background): “Recently, there have been several attempts to build an algebraic foundation of tropical geometry: for instance, [16], [22], [28], [34], [41].”
Logarithmic Gromov–Witten theory with expansions (2019, background): “A careful treatment of extended cone complexes can be found in [1, Section 2], and the relationship with Berkovich spaces and tropical geometry can be found in [60, 68].”
Formes différentielles réelles et courants sur les espaces de Berkovich (2012, unknown): “Voir aussi l’article [48] pour une autre incarnation du lien intime entre géométrie analytique et géométrie tropicale.”
The tropical momentum map: a classification of toric log symplectic manifolds (2014, methodology): “2.19 The notion of tropical domain defined above is closely related to the notion of extended tropicalization Trop(X ) of a complex toric variety X introduced by Kajiwara [10] and Payne [17] (see also [5] for similar ideas).”