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

Abstract

Let $K$ be an algebraically closed, complete, non-Archimedean valued field of characteristic zero, and let $\mathcal{X}$ 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 $\mathcal{X}$ is $K$-analytically special if there exists a connected, finite type algebraic group $G/K$, a dense open subset $\mathcal{U}\subset G^{\text{an}}$ with $\text{codim}(G^{\text{an}}\setminus \mathcal{U})\ge 2$, and an analytic morphism $\mathcal{U}\to \mathcal{X}$ 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.

31 pages; comments welcome!