Kelly Jabbusch: Families over special base manifolds and a conjecture of Campana

Complex varieties are traditionally classified by their Kodiara-Iitaka dimension. Refining the distinction between “general type” and “other”, Campana defined the class of special log varieties (Y, D), characterized by the fact that if A ⊆ Ω^p_Y (log D) is an invertible subsheaf for some p, then κ(A) < p. Generalizing classical Shafarevich Hyperbolicity, he conjectured that any smooth projective family of canonically polarized manifolds over a special base variety is necessarily isotrivial. In this talk I will discuss special pairs and Shafarevich Hyperbolicity, and I will report on joint work with Stefan Kebekus in which we prove Campana’s conjecture for quasi- projective base manifolds of dimension two and three.