Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base Y°, and suppose the family is non-isotrivial. If Y is a smooth compactification of Y°, such that D := Y Y° 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 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 KodairaIitaka 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 Y° is a surface, we see that the non-isotriviality assumption implies that Y° cannot be special in the sense of Campana.
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