Kawamata–Kodaira canonical bundle formula and an application

We discuss Kawamata–Kodaira canonical bundle formula, which is a higher-dimension analogue of Kodaira bundle formula for surfaces. We also introduce a conjecture related by Kawamata and Shokurov. As an application, we shall prove that, if $f\colon X\to Y$ is a smooth surjective morphism between smooth projective varieties over the field of complex numbers, then $-K_X$ is semi-ample implies that $-K_Y$ is semi-ample (it is conjectured by Fujino and Gongyo). This is a joint work with Caucher