Finiteness properties of pseudo-hyperbolic varieties

Abstract

Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem for dynamical systems of infinite order with properties of Prokhorov-Shramov's notion of quasi-minimal models. We also prove a similar result in the geometric setting by using again Amerik's theorem and Prokhorov-Shramov's notion of quasi-minimal model, but also Weil's regularization theorem for birational self-maps and properties of dynamical degrees. Furthermore, in the geometric setting, we obtain an analogue of Kobayashi-Ochiai's finiteness result for varieties of general type, and thereby generalize Noguchi's theorem (formerly Lang's conjecture). Our proof here relies on a deformation-theoretic result for surjective maps of normal varieties due to Hwang-Kebekus-Peternell. Contents 19 8. Dominant self-maps of pseudo-Mordellic varieties 21 9. Dominant self-maps of pseudo-1-bounded varieties 22 References 24

Finiteness of non-constant maps over number fields

“…By Lemma 2.2, for every abelian variety A over K, every rational map A X K factors over Z K . Therefore, since X is two-dimensional (by assumption), it follows that X is of general type (use [JX, Corollary 3.17] and [JX,Lemma 3.23]). This proves the first statement.…”

The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties

“…These results are concerned with endomorphisms of hyperbolic varieties, moduli spaces of maps into a hyperbolic variety, and also the behavior of hyperbolicity in families of varieties. The results presented in these sections are proven in [14,47,48,53,54].…”

“…We will also comment on this more general notion in Section 7. This notion appears in this generality (to our knowledge) for the first time in Vojta's paper [84], and it is also studied in [54]. It is intimately related to the notion of "arithmetic hyperbolicity" [47,51]; see Section 7 for a discussion.…”

“…The notion of grouplessness is well-studied, and sometimes referred to as "algebraic hyperbolicity" or "algebraic Lang hyperbolicity"; see [41], [60, page 176], [57,Remark 3.2.24], or [58,Definition 3.4]. We will only use the term "algebraically hyperbolic" for the notion introduced by Demailly in [28] (see also [14,48,54]). The term "groupless" was first used in [48 Much like Brody hyperbolicity and Mordellicity, grouplessness descends along finite étale morphisms.…”

“…This remark might seem misplaced, but it shows that "grouplessness" as defined above does not capture the non-hyperbolicity of a quasi-projective variety. The "correct" definition in the quasi-projective case is discussed in Section 6 (and is also discussed in [54,84]).…”