Abstract
In this short note, we suggest a definition of almost nonnegativity for orthogonal bisectional curvature and quadratic orthogonal bisectional curvature. Moreover we obtain the differential structure of universal covering of a compact Kähler manifold with almost nonnegative orthogonal bisectional curvature, which implies one of Fang’s conjecture under an additional scalar curvature upper bound condition.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bamler R.H., Cabezas-Rivas E., Wilking B., The Ricci flow under almost non-negative curvature conditions. Invent. Math., 2019, 217(1): 95–126
Chau A., Tam L., On quadratic orthogonal bisectional curvature. J. Differential Geom., 2012, 92(2): 187–200
Cheeger J., Finiteness theorems for Riemannian manifolds. Amer. J. Math., 1970, 92: 61–74
Chen X.X., On Kahler manifolds with positive orthogonal bisectional curvature. Adv. Math., 2007, 215(2): 427–445
Demailly J., Peternell T., Schneider M., Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom., 1994, 3(2): 295–345
Fang F., Kähler manifolds with almost non-negative bisectional curvature. Asian J. Math., 2002, 6(3): 385–398
Gu H., Zhang Z., An extension of Mok’s theorem on the generalized Frankel conjecture. Sci. China Math., 2010, 53(5): 1253–1264
Hamilton R.S., Four-manifolds with positive curvature operator. J. Differential Geom., 1986, 24(2): 153–179
Huang H., A note on Kahler manifolds with almost nonnegative bisectional curvature. Ann. Global Anal. Geom., 2009, 36(3): 323–325
Li Q., Wu D., Zheng F., An example of compact Kähler manifold with nonnegative quadratic orthogonal bisectional curvature. Proc. Amer. Math. Soc., 2013, 141(6): 2117–2126
Mok N., The uniformization theorem for compact Kähler manifolds of nonnegative bisectional curvature. J. Differential Geom., 1988, 27(2): 179–214
Niu Y., A note on nonnegative quadratic orthogonal bisectional curvature. Proc. Amer. Math. Soc., 2014, 142(11): 3975–3979
Petersen P., Tao T., Classification of almost quarter-pinched manifolds. Proc. Amer. Math. Soc., 2009, 137(7): 2437–2440
Acknowledgements
The author is indebted to Prof. Zhenlei Zhang and Liang Cheng for helpful conversations and advices. Part of this paper is carried out when the author was visiting Rutgers University. She would like to thank Prof. Xiaochun Rong for his help and Mathematics Department for their hospitality. Thanks also to China Scholarship Council for supporting the author’s visit. The author also would like to thank the anonymous referee for helpful suggestions. The author was supported by NSFC (No. 11601044) and Beijing Natural Science Foundation (No. Z180004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shao, H. Differential Structure of Kähler Manifold with Almost Nonnegative Orthogonal Bisectional Curvature. Front. Math 18, 743–750 (2023). https://doi.org/10.1007/s11464-017-0024-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-017-0024-x