Finiteness properties of pseudo-hyperbolic varieties

TL;DR

This paper proves finiteness results for rational self-maps of algebraic varieties over number fields and in geometric settings, extending classical conjectures and theorems using advanced dynamical and birational geometry tools.

Contribution

It establishes new finiteness theorems for dominant rational self-maps over number fields and varieties of general type, generalizing prior conjectures and results.

Findings

Finiteness of rational self-maps over number fields with finitely many rational points.

Finiteness results for birational self-maps in geometric settings.

An analogue of Kobayashi-Ochiai's finiteness theorem for varieties of general type.

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).