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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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##### References:
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