  In an earlier work, we investigated some consequences of the existence of a
K\"ahler metric of negative holomorphic sectional curvature on a projective
manifold. In the present work, we extend our results to the case of
semi-negative (i.e., non-positive) holomorphic sectional curvature. In doing
so, we define a new invariant that records the largest codimension of maximal
subspaces in the tangent spaces on which the holomorphic sectional curvature
vanishes. Using this invariant, we establish lower bounds for the nef dimension
and, under certain additional assumptions, for the Kodaira dimension of the
manifold. In dimension two, a precise structure theorem is obtained.
