On almost quasi-negative holomorphic sectional curvature

Yashan Zhang; Tao Zheng

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arXiv · 2010.01314

On almost quasi-negative holomorphic sectional curvature

Yashan Zhang·Tao Zheng

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Abstract

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.

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Yashan Zhang, Tao Zheng. 2020-10-03. On almost quasi-negative holomorphic sectional curvature. https://arxiv.org/abs/2010.01314

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