• Corpus ID: 15635143

The structure of surfaces mapping to the moduli stack of canonically polarized varieties

@article{Kebekus2007TheSO,
  title={The structure of surfaces mapping to the moduli stack of canonically polarized varieties},
  author={Stefan Kebekus and S{\'a}ndor Kov{\'a}cs},
  journal={arXiv: Algebraic Geometry},
  year={2007}
}

• Stefan Kebekus, S. Kovács
• Published 13 July 2007
• Mathematics
• arXiv: Algebraic Geometry

Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a quasi-projective surface that maps to the moduli stack, we employ extension properties of logarithmic pluri-forms to establish a strong relationship between the moduli map and the minimal model program of the surface. As a result, we can describe the fibration induced by the moduli map quite explicitly. A refined affirmative answer to Viehweg's conjecture for families over surfaces follows as a corollary.