Topology of Hybrid Analytifications

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.