A non-Archimedean analogue of Campana's notion of specialness

TL;DR

This paper develops a non-Archimedean analogue of Campana's specialness concept for $K$-analytic spaces, establishing a new framework and proving foundational results that align with classical conjectures.

Contribution

It introduces a novel non-Archimedean definition of specialness for $K$-analytic spaces and verifies its consistency with Campana's classical notion.

Findings

Defines $K$-analytically special spaces using algebraic groups and dense open subsets.

Proves the non-Archimedean analogue captures Campana's specialness.

Establishes auxiliary results on meromorphic maps and hyperbolicity in $K$-analytic geometry.

Abstract

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $\mathscr{X}$ be a $K$-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana's notion of specialness. We say $\mathscr{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $\mathscr{U}\subset G^{\text{an}}$ with $\text{codim}(G^{\text{an}}\setminus \mathscr{U}) \geq 2$, and an analytic morphism $\mathscr{U} \to \mathscr{X}$ which is Zariski dense. With this definition, we prove several results which illustrate that this definition correctly captures Campana's notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between $K$-analytic spaces, the notion of pseudo-$K$-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible $K$-analytic spaces to the analytification of a semi-abelian variety.