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

Abstract

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.

19 pages

References in corpus (1)

• Families of canonically polarized varieties over surfaces

Cited by in corpus (6)

• Differential Forms on Log Canonical Spaces
• Pull-back Morphisms for Reflexive Differential Forms
• Generic positivity and applications to hyperbolicity of moduli spaces
• Differential forms on singular spaces, the minimal model program, and hyperbolicity of moduli stacks
• Uniformisation of higher-dimensional minimal varieties
• Arakelov-Parshin rigidity of towers of curve fibrations