6533b823fe1ef96bd127f60b
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Non-archimedean hyperbolicity and applications
Ariyan JavanpeykarAlberto Vezzanisubject
Abelian varietyPure mathematicsConjectureMathematics - Number TheoryApplied MathematicsGeneral Mathematics010102 general mathematics[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG]Field (mathematics)01 natural sciencesModuli spaceMathematics - Algebraic GeometryMorphism0103 physical sciencesUniformization theoremFOS: MathematicsNumber Theory (math.NT)[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG]010307 mathematical physics0101 mathematicsAbelian groupAlgebraic Geometry (math.AG)Projective varietyMathematicsdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-10-22 | Journal für die reine und angewandte Mathematik (Crelles Journal) |