Hyperbolicity and uniformity of varieties of log general type

TL;DR

This paper explores hyperbolicity properties of quasi-projective varieties with a new positivity condition called almost ample log cotangent bundle, leading to results on subvarieties, integral points, and boundedness related to the Lang-Vojta conjecture.

Contribution

It introduces the notion of almost ample log cotangent bundle for quasi-projective varieties and establishes hyperbolicity and finiteness results extending classical cases.

Findings

Subvarieties of almost ample log cotangent varieties are of log general type.

Varieties with globally generated almost ample log cotangent contain finitely many integral points.

Under the Lang-Vojta conjecture, the number of stably integral points on certain varieties is bounded.

Abstract

Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with almost ample log cotangent sheaf are uniformly bounded.