A non-Archimedean analogue of the Kobayashi semi-distance and its non-degeneracy on Abelian varieties (Q1913514)

The author introduces a natural notion of Kobayashi semi-distance on analytic spaces (analytic in the sense of \textit{V. Berkovich}, `Spectral theory and analytic geometry over non-archimedean fields' (1990; Zbl 0715.14013)] over non archimedean complete and algebraically closed fields \(K\). As in the classical case, it extends the natural distance of the closed unit ball to the affine line and the main question is the following (for the answer over the field of complex numbers, see \textit{R. Brody}, Trans. Am. Math. Soc. 235, 213-219 (1978; Zbl 0416.32013)]; let \(X\) be a projective nonsingular algebraic variety defined over \(K\), viewed as an analytic space over \(K\); does there exist a non constant analytic map \({\mathbf A}^1_K \to X\) if and only if the Kobayashi semi-distance is not a distance? The author shows that, if \(X\) is an abelian variety, the Kobayashi semi-distance on \(X\) is a distance. As he has earlier proved that analytic non constant maps \({\mathbf A}^1_K \to X\), where \(X\) is an abelian variety, do not exist, one has a positive answer to the question, for abelian varieties. The proof uses different methods from those over \(\mathbb{C}\), because the spaces here are not locally compact.