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

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

1999-04-22

arxiv.org/abs/math/9904124v2

Mathematics

Algebraic Geometry

11 pages, LaTeX, we corrected and added some references

Scientific paper

10.1007/s002229900046

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.

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