On almost quasi-negative holomorphic sectional curvature

A recent celebrated theorem of Diverio-Trapani and Wu-Yau states that a compact Kähler manifold admitting a Kähler 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 Kähler 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.

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