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 is false—the log cotangent bundle is never ample. Instead, we define a notion called almost ample that roughly asks that it 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.

The Erdős–Ulam problem, Lang's conjecture and uniformity

“…Analogous statements were proven for (stably) integral points in [1] and [3]; see [4] for a survey of these and related results.…”

Nonspecial varieties and generalised Lang–Vojta conjectures

“…The isotrivial case of Conjecture 7 is known for subvarieties of abelian varieties [Yam15]. For certain cases of Conjecture 7 in the logarithmic setting, see [CZ08,CZ13,Tur17,ADT20,CT19].…”

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

“…Conjecture 1.3 has a wide range of important consequences; we mention for example the abc conjecture of Masser-Oesterlé [33,Conjecture 3], the Bombieri-Lang conjecture [24, Conjecture F.5.2.1] and the Lang-Vojta conjecture [24,Conjecture F.5.3.6]. We refer to [1,2,4,25] for further consequences.…”

“…As the two polynomials are coprime, Bezout's theorem ensures that the number of common solutions is finite and bounded by (deg A) 2 . Hence, α and β are two rational functions in κ(C) independent of u 1 and u 2 , and (3.1) can be rewritten as…”