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<p>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 <i>almost quasi-negative holomorphic sectional curvature</i> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\int _Xc_1(K_X)^n&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∫</mo> <mi>X</mi> </msub> <msub> <mi>c</mi> <mn>1</mn> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>X</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-invariant.</p>

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On Almost Quasi-Negative Holomorphic Sectional Curvature

  • 期刊论文

  • 发表日期：

    2026年5月16日

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  • Yashan Zhang,
  • Tao Zheng

摘要

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 \(\int _Xc_1(K_X)^n>0\) ∫ X c 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. Our proofs make crucial use of Tian’s \(\alpha \) α -invariant.