  Let $(X, \omega_X)$ be a compact K\"ahler manifold such that the
anticanonical bundle $-K_X$ is nef. We prove that the slopes of the
Harder-Narasimhan filtration of the tangent bundle with respect to a
polarization of the form $\omega_X^{n-1}$ are semi-positive. As an application,
we give a characterization of rationally connected compact K\"ahler manifolds
with nef anticanonical bundles. As another application, we give a simple proof
of the surjectivity of the Albanese map.
