We verify some "arithmetic" predictions made by conjectures of Campana, Hassett-Tschinkel, Green-Griffiths, Lang, and Vojta. Firstly, we prove that every dominant endomorphism of an arithmetically hyperbolic variety over an algebraically closed field of characteristic zero is in fact an automorphism of finite order, and that the automorphism group of an arithmetically hyperbolic variety is a locally finite group. To prove these two statements we use (a mild generalization of) a theorem of Amerik on dynamical systems which in turn builds on work of Bell-Ghioca-Tucker, and combine this with a classical result of Bass-Lubotzky. Furthermore, we show that if the automorphism group of a projective variety is torsion, then it is finite. In particular, we obtain that the automorphism group of a projective arithmetically hyperbolic variety is finite, as predicted by Lang's conjectures. Next, we apply this result to verify that projective hyperkahler varieties with Picard rank at least three are 查看全文>>