The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties. (English) Zbl 1208.14027

Let \(f^\circ : X^\circ \rightarrow Y^\circ\) be a family of canonically polarised projective manifolds over a smooth base, i.e. a smooth morphism between quasi-projective manifolds such that the fibres are projective manifolds with ample canonical bundle. The family induces a map from the base \(Y^\circ\) to some moduli stack (of canonically polarised manifolds with fixed Hilbert polynomial) and we define the variation of the family \(f\) as the dimension of the image of this map. If the variation is maximal, i.e. equal to the dimension of \(Y^\circ\), a conjecture of Viehweg claims that the manifold \(Y^\circ\) is of log-general type. This conjecture is classically known to be true if the fibres are curves S. Ju. Arakelov [Math. USSR, Izv. 5(1971), 1277–1302 (1972; Zbl 0248.14004)] or the base \(Y\) is a curve E. Viehweg and K. Zuo [J. Algebr. Geom. 10, No. 4, 781–799 (2001; Zbl 1079.14503)]. More recently the authors dealt with the case where \(Y\) is a surface [Invent. Math. 172, No. 3, 657–682 (2008; Zbl 1140.14031)].
In the paper under review the authors prove a generalised version of Viehweg’s conjecture in the threefold case, i.e. if \(f^\circ : X^\circ \rightarrow Y^\circ\) is a smooth projective family of varieties with semiample canonical bundle and the dimension of the base is at most three, then \(f^\circ\) has maximal variation only if \(Y^\circ\) has log-general type. If the canonical bundle is ample, it is possible to describe the map to the moduli stack: if \((Y, D)\) is a compactification of \(Y^\circ\), any minimal model program for the pair \((Y, D)\) will terminate in a Kodaira or Mori fibre space whose fibration factors the moduli map birationally. Note also that while the authors’ proof of the surface case relied on an explicit discussion of curve arrangements on surfaces and a difficult result by S. Keel and J. McKernan [Rational curves on quasi-projective surfaces. Mem. Am. Math. Soc. 669 (1999; Zbl 0955.14031)], the methods of this paper are more conceptual and might generalise to higher dimension.


14J10 Families, moduli, classification: algebraic theory
14D22 Fine and coarse moduli spaces
Full Text: DOI arXiv


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