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Generalizing the well-known Shafarevich hyperbolicity conjecture, it has beenconjectured by Viehweg that a quasi-projective manifold that admits agenerically finite morphism to the moduli stack of canonically polarizedvarieties is necessarily of log general type. Given a quasi-projective surfacethat maps to the moduli stack, we employ extension properties of logarithmicpluri-forms to establish a strong relationship between the moduli map and theminimal model program of the surface. As a result, we can describe thefibration induced by the moduli map quite explicitly. A refined affirmativeanswer to Viehweg's conjecture for families over surfaces follows as acorollary.

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