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Article Dans Une Revue Journal für die reine und angewandte Mathematik Année : 2021

Non-archimedean hyperbolicity and applications

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Abstract Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K -analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.
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hal-03895759 , version 1 (13-12-2022)

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Ariyan Javanpeykar, Alberto Vezzani. Non-archimedean hyperbolicity and applications. Journal für die reine und angewandte Mathematik, 2021, 778, pp.1-29. ⟨10.1515/crelle-2021-0032⟩. ⟨hal-03895759⟩
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