Topology of Hybrid Analytifications

TL;DR

This paper explores the topological structure of Berkovich analytifications over hybrid fields, revealing conditions under which these spaces are contractible or non-contractible, with implications for number theory and algebraic geometry.

Contribution

It establishes new results on the contractibility of analytifications over hybrid fields, including affine lines, curves, and affine spaces, extending understanding of their topological properties.

Findings

Analytification of affine line or smooth projective curve over countable Archimedean hybrid field is contractible.

Analytification over uncountable hybrid fields can be non-contractible.

Analytification of affine space over non-Archimedean hybrid fields or discrete valuation rings is contractible.

Abstract

We investigate the topological properties of Berkovich analytifications over hybrid fields, that is a field equipped with the maximum of its native norm and the trivial norm. We prove that the analytification of the affine line or of a smooth projective curve over a countable Archimedean hybrid field is contractible, and show that it can be non-contractible when the field is uncountable. Further, we prove that the analytification of affine space over a non-Archimedean hybrid field or over a discrete valuation ring is contractible. As an application, we show that the Berkovich affine line over the ring of integers of a number field is contractible.