A talk in the Bielefeld Arithmetic Geometry Seminar series by Ariyan Javanpeykar from Mainz
A projective variety is arithmetically hyperbolic if it has only finitely many "rational points". What properties should such a variety have? If one believes in conjectures of Green-Griffiths, Lang, and Vojta, such varieties should share many properties in common with varieties of general type and Brody hyperbolic varieties. Motivated by these conjectures, we show that a projective arithmetically hyperbolic variety has only finitely many automorphisms, and
that any surjective endomorphism is an automorphism.