On the Shafarevich conjecture for surfaces of general type over function fields

E. Bedulev, E. Viehweg

Published 1999-04-22, updated 1999-04-30Version 2

For a non-isotrivial family of surfaces of general type over a complex projective curve, we give upper bounds for the degree of the direct images of powers of the relative dualizing sheaf. They imply that, fixing the curve and the possible degeneration locus, the induced morphisms to the moduli scheme of stable surfaces of general type are parameterized by a scheme of finite type. The method extends to families of canonically polarized manifolds, but the modular interpretation requires the existence of relative minimal models.