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).

arXiv comment33 pages. Minor changes to improve exposition. Updated bibliography

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Cited by in corpus (2)

• Demailly's notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps
• The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties