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Analytification is the limit of all tropicalizations. (English) Zbl 1193.14077

The author proves that the analytification of an affine algebraic variety in the sense of V. G. Berkovich [Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs, 33. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013)] is naturally homeomorphic to the inverse limit of tropicalizations of its affine embeddings, and provides an overview of similar results and ideas in the literature. He also defines the tropicalization of a subvariety in a toric variety, gluing it up of tropicalizations of the intersections of the subvariety with the toric orbits. This leads to the quasiprojective version of the first theorem: the analytification of a quasiprojective variety is homeomorphic to the limit of tropicalizations of its quasiprojective embeddings.

MSC:

14T05 Tropical geometry (MSC2010)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32P05 Non-Archimedean analysis
32C15 Complex spaces

Citations:

Zbl 0715.14013
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