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Article Dans Une Revue International Mathematics Research Notices Année : 2022

Finiteness properties of pseudo-hyperbolic varieties

Résumé

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.

Dates et versions

hal-02368901 , version 1 (18-11-2019)

Identifiants

Citer

Ariyan Javanpeykar, Junyi Xie. Finiteness properties of pseudo-hyperbolic varieties. International Mathematics Research Notices, 2022, 3, pp.1601 - 1643. ⟨10.1093/imrn/rnaa168⟩. ⟨hal-02368901⟩
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