Betti realization of varieties defined by formal Laurent series

Achinger, Piotr; Talpo, Mattia

doi:10.2140/gt.2021.25.1919

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Betti realization of varieties defined by formal Laurent series

Piotr Achinger and Mattia Talpo

Geometry & Topology 25 (2021) 1919–1978

DOI: 10.2140/gt.2021.25.1919

Abstract

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

Keywords

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

Mathematical Subject Classification 2010

Primary: 14D06, 14F35, 14F45

References

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

Volume 25, issue 4 (2021)