A study of compatible deformations in non-Archimedean geometry

TL;DR

This paper investigates the existence of deformation retractions compatible with morphisms in non-Archimedean geometry, extending previous work on Berkovich analytifications to more general settings.

Contribution

It adapts Hrushovski--Loeser's methods to establish compatible deformation retractions for morphisms, including the case of relative dimension 1 over smooth curves.

Findings

Compatible deformation retractions exist over a constructible partition of the base.

Proved the existence of compatible retractions for morphisms of relative dimension 1 to smooth curves.

Extended the framework of non-Archimedean deformation retractions to broader classes of morphisms.

Abstract

In 2010, Hrushovski--Loeser showed that the Berkovich analytification of a quasi-projective variety over a non-Archimedean valued field admits a deformation retraction onto a finite simplicial complex. In this article, we adapt the tools and methods developed by Hrushovski--Loeser to study if such deformation retractions can be obtained to be compatible with respect to a given morphism. Amongst other results, we show that compatible deformation retractions exist over a constructible partition of the base and prove the general statement in the case of a morphism of relative dimension 1 where the target is a smooth connected curve.