## Duke Mathematical Journal

### 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 $Y^\circ$ that admits a nonconstant map to the moduli stack, we employ extension properties of logarithmic pluriforms to establish a strong relationship between the moduli map and the minimal model program of $Y^\circ$: 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 often be described quite explicitly. Slightly weaker results are obtained for families of varieties with trivial or more generally semiample canonical bundle.

#### Article information

Source
Duke Math. J. Volume 155, Number 1 (2010), 1-33.

Dates
First available in Project Euclid: 23 September 2010

https://projecteuclid.org/euclid.dmj/1285247217

Digital Object Identifier
doi:10.1215/00127094-2010-049

Mathematical Reviews number (MathSciNet)
MR2730371

Zentralblatt MATH identifier
1208.14027

Subjects
Primary: 14J10: Families, moduli, classification: algebraic theory
Secondary: 14D22: Fine and coarse moduli spaces

#### Citation

Kebekus, Stefan; Kovács, Sándor J. The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. Duke Math. J. 155 (2010), no. 1, 1--33. doi:10.1215/00127094-2010-049. https://projecteuclid.org/euclid.dmj/1285247217.

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