A non-Archimedean Montel's theorem

Abstract : This thesis is devoted to the study of compactness properties of spaces of analytic maps between analytic spaces defined over a non-Archimedean metrized field $k$. We work in the theory of analytic spaces as developed by Berkovich to fully exploit their tame topology. One of our motivations is the strive to introduce a natural notion of Kobayashi hyperbolicity in this setting.We first prove an analogue of Montel’s theorem for analytic maps taking values in a bounded domain of the affine space. In order to do so, we parametrize the space of analytic maps from an open polydisk to a closed one by the analytic spectrum of a suitable Banach $k$-algebra. Our result then follows from the sequential compactness of this space.Our results naturally lead to a definition of normal families, and we subsequently introduce two notions of Fatou sets attached to an endomorphism of the projective space. We show that Fatou components behave like in the complex case and cannot contain non trivial images of the punctured affine line.Thereupon, we apply our normality notion to the study of hyperbolicity in the non-Archimedean setting. We pursue the work of W. Cherry and prove various characterizations of smooth projective varieties whose Cherry-Kobayashi semi distance on the set of rigid points defines the classical topology. We finally obtain a characterization of smooth algebraic curves $X$ of negative Euler characteristic in terms of the normality of certain families of analytic maps taking values in $X$.
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Rita Rodriguez Vazquez. A non-Archimedean Montel's theorem. Algebraic Geometry [math.AG]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLX027⟩. ⟨tel-01618985⟩

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