Positive sheaves of differentials coming from coarse moduli spaces

Jabbusch, Kelly; Kebekus, Stefan

Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2277-2290.

On considère une famille projective lisse de variétés canoniquement polarisées sur une base quasi-projective lisse $Y$. Si la famille n’est pas iso-triviale, Viehweg et Zuo ont montré que toute bonne compactification de $Y$ admet des formes pluricanoniques avec au plus des pôles logarithmiques le long du bord. Plus précisément leur résultat montre qu’une puissance symétrique suffisamment grande du faisceau des différentielles logarithmiques contient un sous-faisceau inversible dont la dimension de Kodaira-Iitaka est au moins égale à la variation de la famille. En suivant la construction de Viehweg-Zuo on montre que le faisceau inversible de Viehweg-Zuo provient, au moins génériquement, de l’espace de module “grossier” associé à la famille.

Comme corollaire immédiat on obtient que la base d’une famille non-isotriviale ne peut pas être spéciale au sens de Campana.

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 $m^{\mathrm{th}}$ symmetric power of this sheaf admits many sections. More precisely, the $m^{\mathrm{th}}$ 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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