  A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a
compact K\"ahler manifold admitting a K\"ahler metric of quasi-negative
holomorphic sectional curvature has an ample canonical line bundle, confirming
a conjecture of Yau. In this paper we shall consider a natural notion of almost
quasi-negative holomorphic sectional curvature and extend this theorem to
compact K\"ahler manifolds of almost quasi-negative holomorphic sectional
curvature. We also obtain a gap-type theorem for the inequality
$\int_Xc_1(K_X)^n>0$ in terms of the holomorphic sectional curvature. In the
discussions, we introduce a capacity notion for the negative part of
holomorphic sectional curvature, which plays a key role in studying the
relation between the almost quasi-negative holomorphic sectional curvature and
ampleness of the canonical line bundle.
