Abstract
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.
Highlights
Conjectures of Green–Griffiths and Lang predict a precise interplay between different notions of hyperbolicity [31, 48]
To motivate the following results, note that the classic Green–Griffiths–Lang conjecture predicts that a projective variety over C is groupless if and only if it is of general type and Brody hyperbolic
We show that K-analytic Brody hyperbolicity can be tested on analytic maps from Gamn and algebraic maps from abelian varieties
Summary
Conjectures of Green–Griffiths and Lang predict a precise interplay between different notions of hyperbolicity [31, 48]. To motivate the following results, note that the classic Green–Griffiths–Lang conjecture predicts that a projective variety over C is groupless if and only if it is of general type and Brody hyperbolic. Part of Lang–Vojta’s conjecture predicts that, if X is a quasi-projective integral scheme over C whose integral subvarieties are of log-general type, X is arithmetically hyperbolic over C [41, Definition 4.1] and Brody hyperbolic It follows from the work of Zuo that every integral subvariety of the moduli space. Motivated by the Lang–Vojta conjecture and the aforementioned properties of the moduli space of abelian varieties, it seems reasonable to suspect that, if K is a complete algebraically closed non-archimedean valued field of characteristic zero, the moduli space XK is K-analytically Brody hyperbolic.
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