Bounds on the denominators in the canonical bundle formula

Abstract

In this work we study the moduli part in the canonical bundle formula of an lc-trivial fibration whose general fibre is a rational curve. If r is the Cartier index of the fibre, it was expected that 12r would provide a bound on the denominators of the moduli part. Here we prove that such a bound cannot even be polynomial in r, we provide a bound N(r) and an example where the smallest integer that clears the denominators of the moduli part is N(r)/r. Moreover we prove that even locally the denominators depend quadratically on r.