Non-archimedean entire curves in closed subvarieties of semi-abelian varieties
- PDF / 247,322 Bytes
- 8 Pages / 439.37 x 666.142 pts Page_size
- 5 Downloads / 166 Views
Mathematische Annalen
Non-archimedean entire curves in closed subvarieties of semi-abelian varieties Jackson S. Morrow1 Received: 31 July 2019 / Revised: 13 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove a non-archimedean analogue of the fact that a closed subvariety of a semiabelian variety is hyperbolic modulo its special locus, and thereby generalize a result of Cherry. Mathematics Subject Classification 32H20 · (32P05)
1 Introduction The Green–Griffiths–Lang–Vojta conjectures predict that a quasi-projective variety X over C is of log-general type if and only if there is a proper closed subset ⊂ X such that X is Brody hyperbolic modulo (i.e., every non-constant holomorphic map C → X (C) factors through (C)); see [3,4,10,18,28,37,41]. For example, this conjecture is known when X is a closed subvariety of an abelian variety by the celebrated theorem of Bloch–Ochiai–Kawamata [9,26,36]. We refer the reader to [34,35] for recent advances. The aim of this paper is to investigate non-archimedean analogues of the Green– Griffiths–Lang–Vojta conjectures. Our starting point is the following theorem, which is the culmination of results in [1,15,16,33,40]. The definitions of the notions appearing in the following theorem are stated in [29, p. 78] and [22, Definitions 7.1, 8.1]. Theorem 1.1 (Abramovich, Faltings, Kawamata, Noguchi, Ueno, Vojta) Let X be a closed subvariety of a semi-abelian variety G over C. Let Sp(X ) be the union of the subvarieties of X which are translates of positive-dimensional closed subgroups of G. Then the following statements hold.
Communicated by Wei Zhang.
B 1
Jackson S. Morrow [email protected] Department of Mathematics, Emory University, Atlanta, GA 30322, USA
123
J. S. Morrow
1. 2. 3. 4.
The subset Sp(X ) is Zariski closed in X . The variety X is of log-general type if and only if Sp(X ) = X . The variety X is arithmetically hyperbolic modulo Sp(X ). The variety X is Brody hyperbolic modulo Sp(X ).
In [2,11–13,30,31,39], the authors investigate possible non-archimedean analogues of the Green–Griffiths–Lang conjecture; however, some of their results contrast the complex analytic setting. Inspired by Cherry’s work, the authors of [24] formulated the “correct” analogue of the Green–Griffiths–Lang conjecture for projective varieties over a non-archimedean valued field K . Our main result is the non-archimedean analogue of the statements (2), (3), and (4) in Theorem 1.1. We refer the reader to Sect. 2 for the definition of a K -analytic Brody hyperbolic variety. Theorem A Let K be an algebraically closed complete non-archimedean valued field of characteristic zero. Let X be a closed subvariety of a semi-abelian variety G over K . Then X is K -analytically Brody hyperbolic modulo Sp(X ). A direct consequence of Theorem A is the following characterization of groupless ([23, Definition 2.1]) closed subvarieties of a semi-abelian variety. Corollary B Let K be an algebraically closed complete non-archimedean valued field of characteristic z
Data Loading...
