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Base spaces of non-isotrivial families of smooth minimal models. (English) Zbl 1006.14004
Bauer, Ingrid (ed.) et al., Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. Berlin: Springer. 279-328 (2002).
The paper under review addresses the question of which complex quasi-projective varieties, say $$U$$, admit a non-constant morphism, $$f$$, to the moduli stack of canonically polarized complex manifolds with fixed Hilbert polynomial, $$h$$. In the case of curves, that is $$\deg h=1$$, there are well known restrictions to $$U$$, or better to a projective non-singular compactification, $$Y$$. The existence of $$f$$ forces $$\Omega_Y^1(\log S)$$ to be big, where $$S=Y\setminus U$$ is a normal crossing divisor. Many papers addressed a similar problem for higher degree $$h$$ [see for instance: L. Migliorini, J. Algebr. Geom. 4, No. 2, 353-361 (1995; Zbl 0834.14021); S. J. Kovács, Math. Ann. 308, No. 2, 347-359 (1997; Zbl 0922.14024); E. Bedulev and E. Viehweg, Invent. Math. 139, No. 3, 603-615 (2000; Zbl 1057.14044)]. In this case the above vector bundle is not necessarily big but still it has some positivity properties.
The main result of the paper is that the existence of a non-constant morphism $$f$$ is only possible if $$U$$ carries multi-differential forms with logarithmic singularities at infinity. The techniques are partly based on previous results by the authors [E. Viehweg and K. Zuo, J. Algebr. Geom. 10, No. 4, 781-799 (2001; Zbl 1079.14503)] were the case of $$U$$ of dimension one is studied, and provide also a vanishing theorem for the space of sections of symmetric powers of logarithmic one forms.
For the entire collection see [Zbl 0989.00069].

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 14D22 Fine and coarse moduli spaces 14F17 Vanishing theorems in algebraic geometry 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli