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The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base. (English) Zbl 1427.14031
Let $$Y$$ be a smooth projective variety and let $$D$$ be a simple normal-crossing reduced divisor. Then $$(Y, D)$$ is called special logarithmic pair if for every invertible subsheaf $$\mathcal L\subseteq \Omega^p_Y \log (D)$$ and every $$p>0$$ the Kodaira dimension $$\kappa(\mathcal L)<p$$. A smooth quasi-projective variety $$Y^\circ$$ is called special if there is a compactification $$Y$$ of $$Y^\circ$$ with a simple normal-crossing boundary divisor $$D$$ such that $$(Y, D)$$ is a special logarithmic pair.
The author proves that every smooth family of canonically polarized manifolds parameterized by a special smooth quasi-projective variety $$Y^\circ$$ is isotrivial.
The paper under review consists of five sections. Section 1 is an introduction. The main result and an overview of the paper are presented here. Some preliminaries on the theory of smooth pairs from [F. Campana, J. Inst. Math. Jussieu 10, No. 4, 809–934 (2011; Zbl 1236.14039)] and [K. Jabbusch and S. Kebekus, Math. Z. 269, No. 3–4, 847–878 (2011; Zbl 1238.14024)] are given in Section 2. In a very short Section 3 an immediate corollary of the result on generic semi-positivity for smooth pairs from [F. Campana and M. Păun, Ann. Inst. Fourier 65, No. 2, 835–861 (2015; Zbl 1338.14012)] is formulated. In Section 4 it is demonstrated that the statement of the main result can be reduced to a certain sufficient condition for a pair $$(X, D)$$ to be of log-general type. Using the corollary from Section 3, the latter is proven in Section 5 by generalizing some of the results from [Zbl 1338.14012].

##### MSC:
 14D22 Fine and coarse moduli spaces 14D23 Stacks and moduli problems 14E30 Minimal model program (Mori theory, extremal rays) 14J10 Families, moduli, classification: algebraic theory 14K10 Algebraic moduli of abelian varieties, classification 14K12 Subvarieties of abelian varieties
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