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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. KovĂˇcs, 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].

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].

Reviewer: Massimiliano Mella (Ferrara)

### 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 |