2009/04/16 by Kelly Jabbusch, Jabbusch, Kelly, Stefan Kebekus +1 · 1 citation
Engineering · Mathematics · Physics and Astronomy · #14D22 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #math.AG #msc:14D22
paper · pdf · doi:10.48550/arxiv.0904.2445
arxiv created 2009/04/16 · openalex publication_date 2009/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.