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Kelly Jabbusch; Stefan Kebekus
Positive sheaves of differentials coming from coarse moduli spaces
(Faisceaux positifs de différentielles provenant d’un espace de modules)
Annales de l'institut Fourier, 61 no. 6 (2011), p. 2277-2290, doi: 10.5802/aif.2673
Article PDF | Reviews MR 2976311 | Zbl 1253.14009
Class. Math.: 14D07, 14D22
Keywords: Moduli space, positivity of differentials

Résumé - Abstract

Consider a smooth projective family of canonically polarized complex manifolds over a smooth quasi-projective complex base $Y^\circ $, and suppose the family is non-isotrivial. If $Y$ is a smooth compactification of $Y^\circ $, such that $D:=Y \setminus Y^\circ $ 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 $m^{\rm th}$ symmetric power of this sheaf admits many sections. More precisely, the $m^{\rm th}$ 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 $Y^\circ $ is a surface, we see that the non-isotriviality assumption implies that $Y^\circ $ cannot be special in the sense of Campana.

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