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2021 Betti realization of varieties defined by formal Laurent series
Piotr Achinger, Mattia Talpo
Geom. Topol. 25(4): 1919-1978 (2021). DOI: 10.2140/gt.2021.25.1919

Abstract

We give two constructions of functorial topological realizations for schemes of finite type over the field ((t)) of formal Laurent series with complex coefficients, with values in the homotopy category of spaces over the circle. The problem of constructing such a realization was stated by D Treumann, motivated by certain questions in mirror symmetry. The first construction uses spreading out and the usual Betti realization over . The second uses generalized semistable models and the log Betti realization defined by Kato and Nakayama, and applies to smooth rigid analytic spaces as well. We provide comparison theorems between the two constructions and relate them to the étale homotopy type and de Rham cohomology. As an illustration of the second construction, we treat two examples, the Tate curve and the nonarchimedean Hopf surface.

Citation

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Piotr Achinger. Mattia Talpo. "Betti realization of varieties defined by formal Laurent series." Geom. Topol. 25 (4) 1919 - 1978, 2021. https://doi.org/10.2140/gt.2021.25.1919

Information

Received: 22 September 2019; Revised: 9 June 2020; Accepted: 13 July 2020; Published: 2021
First available in Project Euclid: 12 October 2021

Digital Object Identifier: 10.2140/gt.2021.25.1919

Subjects:
Primary: 14D06 , 14F35 , 14F45

Keywords: Betti realization , étale homotopy , Kato–Nakayama space , log geometry , rigid analytic space , topology of degenerations

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.25 • No. 4 • 2021
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