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

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

2007-07-13

arxiv.org/abs/0707.2054v1

Mathematics

Algebraic Geometry

19 pages

Scientific paper

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.

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