On Almost Quasi-Negative Holomorphic Sectional Curvature

A recent breakthrough of Wu-Yau [18] proves that a projective manifold admitting a Kähler metric of negative holomorphic sectional curvature has an ample canonical line bundle. Later, it is significantly extended to the case of compact Kähler manifolds by Tosatti-Yang [15], and then further to the case of quasi-negative holomorphic sectional curvature by Wu-Yau [19] and Diverio-Trapani[3]. In this paper, naturally motivated by the Ricci curvature case, we shall consider a notion of almost quasi-negative holomorphic sectional curvature and extend the above-mentioned theorems to compact Kähler manifolds of almost quasi-negative holomorphic sectional curvature. We also obtain a gap-type theorem for the inequality ∫Xc1(KX)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. Our proofs make crucial use of Tian’s α-invariant.