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?
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