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\((\varepsilon ,\delta )\)–Quasi-Negative Curvature and Positivity of the Canonical Bundle

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Abstract

A recent theorem of Diverio–Trapani and Wu–Yau asserts that a compact Kähler manifold with a Kähler metric of quasi-negative holomorphic sectional curvature is projective and canonically polarized. This confirms a long-standing conjecture of Yau. We consider the notion of \((\varepsilon ,\delta )\)–quasi-negativity, generalizing quasi-negativity, and obtain gap-type theorems for \(\int _X c_1(K_X)^n>0\) in terms of the real bisectional curvature and weighted orthogonal Ricci curvature. These theorems are also a generalization of that results by Zhang-Zheng (arXiv:2010.01314v4 ) and Chu-Lee-Tam (Trans Am Math Soc 375(11):7925–7944, 2022).

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Acknowledgements

The first named author would like to thank his former Ph.D. advisors Ben Andrews and Gang Tian for their support and encouragement. He would also like to thank Simone Diverio, Jeffrey Streets, Finnur Lárusson, Ramiro Lafuente, and James Stanfield for many valuable discussions and communications. The authors would like to thank Yashan Zhang for his support, and for many invaluable discussions and suggestions.

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Authors and Affiliations

  1. The University of Queensland, St. Lucia, QLD, 4067, Australia

    Kyle Broder

  2. School of Mathematical Sciences, Zhejiang Normal University, Jinhua, 321004, Zhejiang, China

    Kai Tang

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  1. Kyle Broder
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Correspondence to Kai Tang.

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The first named author was supported by the Australian Government through the Australian Research Council’s Discovery Projects funding scheme (project DP220102530). The second named author was supported by National Natural Science Foundation of China (Grant No.12001490).

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Broder, K., Tang, K. \((\varepsilon ,\delta )\)–Quasi-Negative Curvature and Positivity of the Canonical Bundle. J Geom Anal 34, 180 (2024). https://doi.org/10.1007/s12220-024-01619-4

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