Families of canonically polarized varieties over surfaces

Details Families of canonically polarized varieties over surfaces Families of canonically polarized varieties over surfaces

2005-11-15

arxiv.org/abs/math/0511378v4

Mathematics

Algebraic Geometry

final version, to appear in Invent. Math. A rather minor issue in construction 6.10 and a typo has been fixed

Scientific paper

Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by Viehweg that the base of a smooth family of canonically polarized varieties is of log general type if the family is of maximal variation. In this paper, we relate the variation of a family to the logarithmic Kodaira dimension of the base and give an affirmative answer to Viehweg's conjecture for families parametrized by surfaces.

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