Inventiones mathematicae

, Volume 172, Issue 3, pp 657–682 | Cite as

Families of canonically polarized varieties over surfaces



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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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

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