Images of manifolds with semi-ample anti-canonical divisor (Q2796770)

Let \(X\) be a smooth complex projective variety; a divisor \(D\) on \(X\) is called semiample if, for some positive integer \(m\) the linear system \(|mD|\) is base point free.NEWLINENEWLINEIn the paper under review the authors prove that -- as conjectured by Fujino and Gongyo -- if the anticanonical bundle of \(X\) is semiample and \(f:X \to Z\) is a smooth morphism, then also the anticanonical bundle of \(Z\) is semiample.NEWLINENEWLINESimilar statements were known to hold replacing ``semiample'' with ``nef'', ``ample'' or ``nef and big''. As shown by different counterexamples the smoothness assumption is essential.NEWLINENEWLINEThe proof employs on advanced tools of the Minimal Model Program.