#### Volume 25, issue 4 (2021)

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Betti realization of varieties defined by formal Laurent series

### Piotr Achinger and Mattia Talpo

Geometry & Topology 25 (2021) 1919–1978
##### Abstract

We give two constructions of functorial topological realizations for schemes of finite type over the field $ℂ\left(\left(t\right)\right)$ 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 $ℂ$. 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.

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##### Keywords
Kato–Nakayama space, log geometry, Betti realization, rigid analytic space, topology of degenerations, étale homotopy
##### Mathematical Subject Classification 2010
Primary: 14D06, 14F35, 14F45
##### Publication
Received: 22 September 2019
Revised: 9 June 2020
Accepted: 13 July 2020
Published: 12 July 2021
Proposed: Dan Abramovich
Seconded: Haynes R Miller, Paul Seidel
##### Authors
 Piotr Achinger Instytut Matematyczny Polskiej Akademii Nauk Warsaw Poland Mattia Talpo Dipartimento di Matematica Università di Pisa Pisa Italy