Let $(X, omega)$ be a compact Kähler manifold. The canonical bundle $K_X$ is said to be semi-ample if a sufficiently high power of it furnishes a holomorphic map $f : X to Y$ onto a normal projective variety $Y$, with the dimension of $Y$ being the Kodaira dimension of $X$. To those in the establishment, $Y$ is the canonical model of $X$. By adjunction, one can show that $f : X to Y$ is a Calabi-Yau fiber space, i.e., $f$ is a proper surjective holomorphic map whose smooth fibers have vanishing first Chern class. In particular, compact Kähler manifolds can be constructed by considering such Calabi-Yau fiber spaces.
I would like to collect as many explicit examples of such objects, preferably with $dim(Y) > 1$ and $f$ not being an elliptic fibration.