A complex manifold \(X\) is said to be hyperbolic (in the sense of Brody) if every analytic map from \({\mathbb C}\) to \(X\) is constant. Kobayashi and Zaidenberg have conjectured that the complement of a generic hypersurface in \({\mathbb P}^n\) with degree at least \(2n+1\) is hyperbolic.
The paper under review deals with the non-Archimedean analogue of this conjecture (which is straightforward except that the degree \(2n+1\) has to be replaced by \(2n\)). In this direction, the authors prove that \({\mathbb P}^n \setminus \bigcup_{i=1}^n D_{i}\) is hyperbolic if the \(D_{i}\)’s are nonsingular hypersurfaces of degree at least \(2\) that intersect transversally (theorem \(3\)). For \({\mathbb P}^2\), they get a better result: they only require that the sum of the degrees be at least \(4\) (theorem \(4\)). The proofs mainly rely on the non-Archimedean Picard theorem: an irreducible projective curve with two points omited is hyperbolic [see V. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33 (1990; Zbl 0715.14013)].
Reviewer’s remark: The paper is well-written and easy to read. Still, we regret that the general setting is not made clear (Berkovich spaces presumably) and that the authors stick to the case of an algebraically closed non-Archimedean field with a non-trivial valuation (the results easily extend to a more general situation).