Non-archimedean hyperbolicity and applications

TL;DR

This paper introduces a new non-archimedean hyperbolicity concept for rigid analytic varieties, demonstrating its implications for algebraic geometry, including properties of moduli spaces and analogues of classical theorems in a non-archimedean setting.

Contribution

It develops a novel notion of Brody hyperbolicity in non-archimedean geometry and applies it to prove new results about abelian varieties and moduli spaces.

Findings

Moduli space of abelian varieties is $K$-analytically Brody hyperbolic.

Specializations of varieties preserve the existence of non-constant morphisms from abelian varieties.

The moduli space satisfies a non-archimedean analogue of the Theorem of the Fixed Part.

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.