1. Hyperbolicity and Uniformity of Varieties of Log General type
- Author
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Amos Turchet, Kristin DeVleming, Kenneth Ascher, Ascher, Kenneth, Devleming, Kristin, and Turchet, Amos
- Subjects
Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Generalization ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Sheaf ,Trigonometric functions ,Uniform boundedness ,Cotangent bundle ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics - 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., v5: exposition greatly improved. Previous section on function fields removed, to be expanded upon in a future paper. To appear in IMRN
- Published
- 2020