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Hyperbolicity notions for varieties defined over a non-Archimedean field. (English) Zbl 1440.32009
Summary: Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semidistance \(d_{\text{CK}} \), which he introduced for analytic spaces defined over a non-Archimedean metrized field \(k\). We prove various characterizations of smooth projective varieties for which \(d_{\text{CK}}\) is an actual distance.
Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve \(X\) defined over \(k\). We prove a non-Archimedean analogue of the equivalence between having a negative Euler characteristic and the normality of certain families of analytic maps taking values in \(X\).

MSC:
32P05 Non-Archimedean analysis
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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