The structure of surfaces and threefolds 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 threefold Y that admits a non-constant map 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 Y: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary. For families over surfaces, the moduli map can be often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial, or more generally semi-ample canonical bundle.

References in corpus (1)

• Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties

Cited by in corpus (9)

• Differential Forms on Log Canonical Spaces
• Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of Abelian varieties
• Pull-back Morphisms for Reflexive Differential Forms
• Viehweg's hyperbolicity conjecture is true over compact bases
• The isotriviality of families of canonically-polarized manifolds over a special quasi-projective base
• Brody hyperbolicity of base spaces of certain families of varieties
• Boundedness results for families of non-canonically polarized projective varieties
• Families of canonically polarized manifolds over log Fano varieties
• Logarithmic base change theorem and smooth descent of positivity of log canonical divisor