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

TL;DR

This paper investigates the structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties, proving Viehweg's conjecture for threefolds and providing explicit descriptions for families over surfaces.

Contribution

It establishes a strong link between the moduli map and the minimal model program for threefolds, confirming Viehweg's conjecture in this case and analyzing families over surfaces.

Findings

Viehweg's conjecture holds for threefolds with non-constant moduli maps.

Explicit descriptions of moduli maps over surfaces are provided.

Results extend to families with semi-ample canonical bundles.

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.