  In this paper, with the aim of establishing a structure theorem for a compact
K\"ahler manifold $X$ with semi-positive holomorphic sectional curvature, we
study a morphism $\phi: X \to Y$ to a compact K\"ahler manifold $Y$ with
pseudo-effective canonical bundle. We prove that the morphism $\phi$ is always
smooth (that is, a submersion), the image $Y$ admits a finite etale cover $T
\to Y$ by a complex torus $T$, and further that all the fibers are isomorphic
when $X$ is projective. Moreover, by applying a modified method to maximal
rationally connected fibrations, we show that $X$ is rationally connected, if
$X$ is projective and $X$ has no truly flat tangent vectors at some point
(which is satisfied when the holomorphic sectional curvature is
quasi-positive). This result gives a generalization of Yau's conjecture. As a
further application, we obtain a uniformization theorem for compact K\"ahler
surfaces with semi-positive holomorphic sectional curvature.
