  On a compact K\"ahler manifold, we introduce a notion of almost nonpositivity
for the holomorphic sectional curvature, which by definition is weaker than the
existence of a K\"ahler metric with semi-negative holomorphic sectional
curvature. We prove that a compact K\"ahler manifold of almost nonpositive
holomorphic sectional curvature has a nef canonical line bundle, contains no
rational curves and satisfies some Miyaoka-Yau type inequalities. In the course
of the discussions, we attach a real value to any fixed K\"ahler class which,
up to a constant factor depending only on the dimension of manifold, turns out
to be an upper bound for the nef threshold.
