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arXiv · 1808.09880

Non-archimedean hyperbolicity and applications

Ariyan Javanpeykar·Alberto Vezzani
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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 zero. 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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Ariyan Javanpeykar, Alberto Vezzani. 2018-08-29. Non-archimedean hyperbolicity and applications. https://doi.org/10.1515/crelle-2021-0032

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