Finiteness properties of pseudo-hyperbolic varieties

Ariyan Javanpeykar; Junyi Xie

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arXiv · 1909.12187

Finiteness properties of pseudo-hyperbolic varieties

Ariyan Javanpeykar·Junyi Xie

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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).

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BibTeX RIS

Ariyan Javanpeykar, Junyi Xie. 2019-09-26. Finiteness properties of pseudo-hyperbolic varieties. https://arxiv.org/abs/1909.12187

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