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Families of canonically polarized manifolds over log Fano varieties
- Part of
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- Journal:
- Compositio Mathematica / Volume 149 / Issue 6 / June 2013
- Published online by Cambridge University Press:
- 26 March 2013, pp. 1019-1040
- Print publication:
- June 2013
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- Article
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Let
$(X,D)$ be a dlt pair, where 
$X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over 
$X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor 
$-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a 
$\mathbb Q$-factorialization of 
$X$. As 
$\mathbb Q$-factorializations are generally not unique, we use flops to pass from one 
$\mathbb Q$-factorialization to another, proving the existence of a 
$\mathbb Q$-factorialization suitable for our purposes.
