Positive sheaves of differentials coming from coarse moduli spaces

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base U, and suppose the family is non-isotrivial. If Y is a smooth compactification of U, such that D := Y U is a simple normal crossing divisor, then we can consider the sheaf of differentials with logarithmic poles along D. Viehweg and Zuo have shown that for some number m>0, the m-th symmetric power of this sheaf admits many sections. More precisely, the m-th symmetric power contains an invertible sheaf whose Kodaira-Iitaka dimension is at least the variation of the family. We refine this result and show that this “Viehweg-Zuo sheaf” comes from the coarse moduli space associated to the given family, at least generically. As an immediate corollary, if U is a surface, we see that the non-isotriviality assumption implies that U cannot be special in the sense of Campana.