IMAGES OF MANIFOLDS WITH SEMI-AMPLE ANTI-CANONICAL DIVISOR

We prove that if f : X -> Z is a smooth surjective morphism between projective manifolds and if -K-X is semi-ample, then -K-Z is also semiample. This was conjectured by Fujino and Gongyo. We list several counterexamples to show that this fails without the smoothness assumption on f. We prove the above result by proving some results concerning the moduli divisor of the canonical bundle formula associated to a klt-trivial fibration (X, B)-> Z.