Families over special base manifolds and a conjecture of Campana

Abstract

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Géométriques. We discuss a weak variant of the Harder–Narasimhan Filtration and prove a version of the Bogomolov–Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana’s theory is included for the reader’s convenience.