Overconvergent subanalytic subsets in the framework of Berkovich spaces

TL;DR

This paper investigates overconvergent subanalytic subsets in non-archimedean Berkovich spaces, analyzing their local properties, topological behavior, and providing corrections to previous results, especially for spaces of dimension two.

Contribution

It introduces a detailed study of overconvergent subanalytic subsets, clarifies their behavior under Berkovich topology, and corrects earlier misconceptions, particularly in dimension two.

Findings

Overconvergent subanalytic sets behave well with Berkovich topology.

They do not behave well with the G-topology, leading to counter-examples.

A simplified characterization is achieved for dimension two spaces.

Abstract

We study the class of overconvergent subanalytic subsets of a $k$-affinoid space $X$ when $k$ is a non-archimedean field. These are the images along the projection $X \times B^n \to X$ of subsets defined with inequalities between functions of $X\times B^n$ which are overconvergent in the variables of $B^n$. In particular, we study the local nature, with respect to $X$, of overconvergent subanalytic subsets. We show that they behave well with respect to the Berkovich topology, but not to the $G$-topology. This gives counter-examples to previous results on the subject, and a way to correct them. Moreover, we study the case dim$(X)=2$, for which a simpler characterisation of overconvergent subanalytic subsets is proven.