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Families of canonically polarized manifolds over log Fano varieties

  • Daniel Lohmann (a1)

Abstract

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.

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References

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Families of canonically polarized manifolds over log Fano varieties

  • Daniel Lohmann (a1)

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