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Projective varieties with ample cotangent bundle satisfy many notions ofhyperbolicity, and one goal of this paper is to discuss generalizations toquasi-projective varieties. A major hurdle is that the naive generalizationfails, i.e. the log cotangent bundle is never ample. Instead, we define anotion called almost ample which roughly asks that the log cotangent is aspositive as possible. We show that all subvarieties of a quasi-projectivevariety with almost ample log cotangent bundle are of log general type. Inaddition, if one assumes globally generated then we obtain that such varietiescontain finitely many integral points. In another direction, we show that theLang-Vojta conjecture implies the number of stably integral points on curves oflog general type, and surfaces of log general type with almost ample logcotangent sheaf are uniformly bounded.

Hyperbolicity and uniformity of varieties of log general type