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            <title xml:lang="en">Arithmetic hyperbolicity: endomorphisms, automorphisms, hyperkähler varieties, geometricity</title>
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                <forename type="first">Ariyan</forename>
                <surname>Javanpeykar</surname>
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                <title xml:lang="en">Arithmetic hyperbolicity: endomorphisms, automorphisms, hyperkähler varieties, geometricity</title>
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                    <forename type="first">Ariyan</forename>
                    <surname>Javanpeykar</surname>
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              <p>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 auto-morphism 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 hyperkähler varieties with Picard rank at least three are not arithmetically hyperbolic. Finally, we show that arithmetic hyperbolicity is a "geometric" notion, as predicted by Green-Griffiths-Lang's conjecture, under suitable assumptions related to Demailly's notion of algebraic hyperbolicity. For instance, if k is an algebraically closed subfield of the field of complex numbers and X is an arithmetically hyperbolic variety over k such that X is Brody hyperbolic over the complex numbers, then X remains arithmetically hyperbolic after any algebraically closed field extension of k.</p>
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