Families over special base manifolds and a conjecture of Campana

Details Families over special base manifolds and a conjecture of Campana Families over special base manifolds and a conjecture of Campana

2009-05-12

arxiv.org/abs/0905.1746v2

Mathematische Zeitschrift, Volume 269, Issue 3 (2011), Page 847-878

Mathematics

Algebraic Geometry

Final version, to appear in Math. Zeitschrift

Scientific paper

10.1007/s00209-010-0758-6

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective 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 Geometriques. 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 self-contained introduction to Campana's theory is included for the reader's convenience.

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