Given an algebraic fiber space $X \to B$ where $X$ and $B$ are smooth projective varieties over $\mathbb{C}$, it is known that the Kodaira dimensions satisfy the following subadditivity property: $$\kappa(X) \ge \kappa(\text{general fiber}) + \kappa(B).$$ Does the same property holds for compact Kähler manifolds? If it is still unknown, are there any partial results in this direction?

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That statement is not known in general for projective varieties: that is the Iitaka conjecture. – user5117 Mar 19 '13 at 19:38
By the way, if you're interested in knowing how much is known about the Iitaka conjecture in the projective setting, one reference is the following paper by Birkar: dpmms.cam.ac.uk/~cb496/iitaka-6.pdf. There, he shows that it is true when dim X (in your notation) is 6 or less, and also if dim B is 2 and the general fibre F has Kodaira dimension 0. – user5117 Mar 20 '13 at 16:49
If $X$ is just a compact complex manifold (not Kähler), then the statement is false. There is an example in the book of K. Ueno, "Classification theory of algebraic varieties and compact complex spaces", see the reference on page 1 of this paper arxiv.org/abs/1204.3165. I am not aware of any Kähler counterexample. – YangMills Mar 21 '13 at 3:41