Hyperbolicity notions for varieties defined over a non-Archimedean field

TL;DR

This paper investigates hyperbolicity concepts for varieties over non-Archimedean fields, extending classical complex geometry notions and characterizing when certain metrics are genuine distances.

Contribution

It provides new characterizations of hyperbolicity and distance properties for varieties over non-Archimedean fields, building on W. Cherry's work.

Findings

Characterized varieties where the Kobayashi semi-distance is a true metric.

Established non-Archimedean analogues of classical hyperbolicity results.

Linked negative Euler characteristic with normality of analytic map families.

Abstract

Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semi distance dCK that he introduced for analytic spaces defined over a non-Archimedean metrized field k. We prove various characterizations of smooth projective varieties for which dCK is an actual distance. Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve X defined over k. We prove a non-Archimedean analogue of the equivalence between having negative Euler characteristic and the normality of certain families of analytic maps taking values in X.