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Title of Journal: Math Z

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Abbravation: Mathematische Zeitschrift

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Springer-Verlag

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10.1016/0165-1765(95)00796-2

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1432-1823

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Families over special base manifolds and a conject

Authors: Kelly Jabbusch Stefan Kebekus
Publish Date: 2010/08/06
Volume: 269, Issue: 3-4, Pages: 847-878
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Abstract

Consider a smooth projective family of canonically polarized varieties over a smooth quasiprojective base manifold Y all defined over the complex numbers It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana We prove the conjecture when Y is a surface or threefold The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces as introduced by Campana in his theory of Orbifoldes Géométriques We discuss a weak variant of the Harder–Narasimhan Filtration and prove a version of the Bogomolov–Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account A brief but selfcontained introduction to Campana’s theory is included for the reader’s convenienceK Jabbusch and S Kebekus were supported by the DFGForschergruppe “Classification of Algebraic Surfaces and Compact Complex Manifolds” in full and in part respectively The work on this paper was finished while the authors visited the 2009 Special Year in Algebraic Geometry at the Mathematical Sciences Research Institute Berkeley Both authors would like to thank the MSRI for support

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