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 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 that X is K-analytically special if there exist a connected, finite-type algebraic group G/K, a dense open subset U ⊂ G an with codim(G an \ U ) ⩾ 2, and an analytic morphism U → 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.