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

Abstract

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.

11 pages, LaTeX, we corrected and added some references

References in corpus (2)

• Minimal models and boundedness of stable varieties
• Deformations of canonical singularities

Cited by in corpus (16)

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• Shafarevich's Conjecture for CY Manifolds I
• Ample Divisors, Automorphic Forms and Shafarevich's Conjecture
• Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin boundedness for singular varieties
• Discreteness of minimal models of Kodaira dimension zero and subvarieties of moduli stacks
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• Boundedness results for families of non-canonically polarized projective varieties
• Arakelov (in)equalities
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• On uniformly effective birationality and the Shafarevich Conjecture over curves
• Arakelov-Parshin rigidity of towers of curve fibrations