Hyperbolicity Notions for Varieties Defined over a Non-Archimedean Field

Abstract

Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semidistance $d_{\mathrm{CK}}$, which he introduced for analytic spaces defined over a non-Archimedean metrized field $k$. We prove various characterizations of smooth projective varieties for which $d_{\mathrm{CK}}$ 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 a negative Euler characteristic and the normality of certain families of analytic maps taking values in $X$.