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In this paper, with the aim of establishing a structure theorem for a compactK\"ahler manifold $X$ with semi-positive holomorphic sectional curvature, westudy a morphism $\phi: X \to Y$ to a compact K\"ahler manifold $Y$ withpseudo-effective canonical bundle. We prove that the morphism $\phi$ is alwayssmooth (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 isomorphicwhen $X$ is projective. Moreover, by applying a modified method to maximalrationally 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 isquasi-positive). This result gives a generalization of Yau's conjecture. As afurther application, we obtain a uniformization theorem for compact K\"ahlersurfaces with semi-positive holomorphic sectional curvature.

On morphisms of compact K\"ahler manifolds with semi-positive holomorphic sectional curvature