  In this paper, we establish a "pseudo-effective" version of the holonomy
principle for compact K\"{a}hler manifolds with nonnegative holomorphic
sectional curvature. As applications, we prove that if a compact complex
manifold $M$ admits a K\"{a}hler metric $\omega$ with nonnegative holomorphic
sectional curvature and $(M,\omega)$ has no nonzero truly flat tangent vector
at some point (which is satisfied when the holomorphic sectional curvature is
quasi-positive), then $M$ must be projective and rationally connected. This
answers a problem raised by Matsumura and Yang and extends Yau's conjecture. We
also prove that a compact simply connected K\"{a}hler manifold with nonnegative
holomorphic sectional curvature is projective and rationally connected.
Additionally, we classify non-projective K\"{a}hler 3-dimensional manifolds
with nonnegative holomorphic sectional curvature. Furthermore, we show that a
compact K\"{a}hler manifold admits a Hermitian metric with positive real
bisectional curvature is a projective and rationally connected manifold.
