Nonspecial varieties and Generalized Lang-Vojta conjectures

Abstract

We construct a family of fibered threefolds Xm → (S, ∆) that are weakly special but not special in the sense of Campana. We prove that if m is big enough, the threefolds Xm present behaviours that contradict the function field and analytic analogues of a conjecture of Abramovich and Colliot-Thélène, thus providing evidences towards Campana's conjecture. This is obtained using an orbifold analogue of [CZ10, Main Theorem] in the function field and analytic setting, which we prove adapting the recent method of Ru and Vojta in [RV]. We also formulate some generalizations of known conjectures on exceptional loci that fit into Campana's program and prove some cases over function fields.

The Ru-Vojta result for subvarieties

“…, 1) be the standard basis of R r , and let t be as in (11). Then, by convexity of F and by (10), we get…”

Hyperbolicity and Specialness of Symmetric Powers

“…It has been shown in [CP07] and [RTJ20] that one cannot replace "special" by "weakly-special" in Conjecture 3.7 in the analytic and function fields settings.…”