Betti realization of varieties defined by formal Laurent series

TL;DR

This paper develops two functorial topological realization methods for schemes over formal Laurent series fields, connecting algebraic geometry, topology, and mirror symmetry, with applications to examples like Tate curves.

Contribution

It introduces two new constructions of topological realizations for schemes over $ ext{C}( ext{(t)})$, linking them to existing theories and providing comparison theorems.

Findings

Two constructions of topological realizations are established.

Comparison theorems relate the constructions to étale homotopy and de Rham cohomology.

Applications include analysis of Tate curve and non-archimedean Hopf surface.

Abstract

We give two constructions of functorial topological realizations for schemes of finite type over the field $\mathbb{C}(\!(t)\!)$ 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 $\mathbb{C}$. The second uses generalized semistable models and 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 \'etale homotopy type and de Rham cohomology. As an illustration of the second construction, we treat two examples, the Tate curve and the non-archimedean Hopf surface.