Families of canonically polarized manifolds over log Fano varieties

TL;DR

This paper proves that smooth families of canonically polarized varieties over certain log Fano pairs are isotrivial, extending previous results and employing the minimal model program with Q-factorializations and flops.

Contribution

It extends the isotriviality result to log Fano pairs with dlt singularities using advanced MMP techniques and Q-factorializations.

Findings

Any extremal ray of the moving cone is generated by a family of curves.

These curves are contracted after a run of the minimal model program.

The result generalizes Araujo's theorem from klt to dlt pairs.

Abstract

Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X\S is isotrivial if the divisor -(K+D) is ample. This result extends results of Viehweg-Zuo and Kebekus-Kovacs. 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 Q-factorialization of X. As Q-factorializations are generally not unique, we use flops to pass from one Q-factorialization to another, proving the existence of a Q-factorialization suitable for our purposes.