I am looking for a proof of the following statement:


Let $f: X \to B$ be a surjective morphism between smooth projective varieties
such that $-K\_X$ is nef and $B$ is non-uniruled then Kodaira dimension of base $\kappa(B)= 0$. What about when we replace projective varieties with "Kähler manifolds"



Kodaira dimension of the base
I think you are looking for [Q. Zhang, On projective varieties with nef anticanonical divisors, Math. Ann. 332 (2005), 697–703.]


See also [Meng Chen; Qi Zhang. On a question of Demailly-Peternell-Schneider. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1853–1858] for a generalization to tell you that in fact $K\_B\sim\_{\mathbb{Q}}0$.


For Kahler case, I believe Junyan Cao has some work. Just look at his papers. 


