On almost quasi-negative holomorphic sectional curvature

TL;DR

This paper extends the Yau conjecture-related theorem to manifolds with almost quasi-negative holomorphic sectional curvature, establishing new links between curvature conditions and the ampleness of the canonical bundle.

Contribution

It introduces the concept of almost quasi-negative holomorphic sectional curvature and generalizes existing theorems to this broader setting, including a gap theorem involving the canonical bundle.

Findings

Extension of the Yau conjecture theorem to almost quasi-negative curvature

Introduction of a capacity notion for negative curvature parts

A gap theorem relating curvature to the positivity of the canonical bundle

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.