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arXiv · 1207.4070

On the moduli part of the Kawamata-Kodaira canonical bundle formula

Caucher Birkar·Yifei Chen
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Abstract

It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration $(X,B)\to Z$ is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking comes from a section of $K_X+B$. We apply the above result to settle a conjecture of Fujino and Gongyo: if $f\colon X\to Z$ is a smooth surjective morphism of smooth projective varieties with $-K_X$ semi-ample, then $-K_Z$ is also semi-ample. We list several counter-examples to show that this fails without the smoothness assumption on $f$.

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Caucher Birkar, Yifei Chen. 2012-07-17. On the moduli part of the Kawamata-Kodaira canonical bundle formula. https://arxiv.org/abs/1207.4070

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