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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"

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  • $\begingroup$ Zhang showed:Let $X$ be a normal projective variety and $D$ an effective $\mathbb Q$-divisor on $X$ such that the pair $(X,D)$ is log canonical and $−(K_X+D)$ is nef. Let $ f: X\to Y$ be a surjective morphism onto a normal variety $Y $ such that $−K_Y$ is $\mathbb Q$-Cartier. Then $-K_Y$ is pseudo-effective. $\endgroup$
    – 1984
    Commented Jan 1, 2018 at 3:27

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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.

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    $\begingroup$ For the Kahler case I don't think this is known yet (Cao's results mostly concern the structure of the Albanese map of $X$). As a warning sign that this may not be easy, the algebraic results that you quote use BDPP's theorem, which is still open in the Kahler case (in dimensions higher than $3$) $\endgroup$
    – YangMills
    Commented Dec 30, 2017 at 9:35
  • $\begingroup$ @YangMills, Interesting comment $\endgroup$
    – 1984
    Commented Dec 30, 2017 at 19:25

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