Hyperbolicity and uniformity of varieties of log general type

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.

arXiv commentv5: exposition greatly improved. Previous section on function fields removed, to be expanded upon in a future paper. To appear in IMRN

References in corpus (6)

• Differential Forms on Log Canonical Spaces
• Noetherian approximation of algebraic spaces and stacks
• The canonical sheaf of Du Bois singularities
• Uniform bounds for the number of rational points on curves of small Mordell--Weil rank
• A high fibered power of a family of varieties of general type dominates a variety of general type
• A fibered power theorem for pairs of log general type

Cited by in corpus (3)

• Nonspecial varieties and Generalized Lang-Vojta conjectures
• The Erdős-Ulam problem, Lang's conjecture, and uniformity
• Hyperbolicity of smooth logarithmic and orbifold pairs in ${\mathbb{P}}^n$