Abstract
Motivated by the recent work of Chu–Lee–Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive k-Ricci curvature, we consider a natural notion of almost nonpositive k-Ricci curvature, which is weaker than the existence of a Kähler metric with nonpositive k-Ricci curvature. When \(k=1\), this is just the almost nonpositive holomorphic sectional curvature introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with k-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact Kähler manifold of almost nonpositive k-Ricci curvature must have nef canonical line bundle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chu, J.C., Lee, M.C., Tam, L.F.: Kähler manifolds with negative \(k\)-Ricci Curvature. arXiv:2009.06297 (2020)
Diverio, S., Trapani, S.: Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle. J. Differ. Geom. 111(2), 303–304 (2019)
Guedj, V., Zeriahi, A.: Regularizing properties of the twisted Kahler-Ricci flow. J. Reine Angew. Math. 729, 275–304 (2017)
Heier, G., Lu, S., Wong, B.: Kähler manifolds of semi-negative holomorphic sectional curvature. J. Differ. Geom. 104, 419–441 (2016)
Heier, G., Wong, B.: On projective Kähler manifolds of partially positive curvature and rational connectedness. arXiv:1509.02149 (2015)
Hitchin, N.: On the curvature of rational surfaces. In Differential Geometry (Proc. Sympos. Pure Math., Vol XXVII, Part 2, Stanford University, Stanford, Calif., 1973), pp. 65–80. American Mathematical Society, Providence, RI (1975)
Lee, M.C., Streets, J.: Complex manifolds with negative curvature operator. Int. Math. Res. Not. arXiv:1903.12645 (2019)
Li, C., Ni, L., Zhu, X.H.: An application of \(C^{2}\)-estimate for a complex Monge–Ampère equation. arXiv:2011.13508 (2020)
Ni, L.: Liouville theorems and a Schwarz Lemma for holomorphic mappings between Kähler manifolds. Commun. Pure Appl. Math. 74, 1100–1126 (2021)
Ni, L.: The fundamental group, rational connectedness and the positivity of Kähler manifolds. J. Reine Angew. Math. (Crelle) 774, 267–299 (2021)
Ni, L., Zheng, F.Y.: Comparison and vanishing theorems for Kähler manifolds. Calc. Var. 57, 151 (2018). https://doi.org/10.1007/s00526-018-1431-x
Ni, L., Zheng, F.Y.: Positivity and Kodaira embedding theorem. ArXiv (2018). arXiv:1804.09609
Nomura, R.: Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach. Int. Math. Res. Not. IMRN 21, 6611–6616 (2018)
Royden, H.L.: The Ahlfors–Schwarz lemma in several complex variables. Commentarii Mathematici Helvetici 55, 547–558 (1980)
Tang, K.: On real bisectional curvature and Kähler-Ricci flow. Proc. Am. Math. Soc. 147(2), 793–798 (2019)
Tosatti, V., Yang, X.K.: An extension of a theorem of Wu–Yau. J. Differ. Geom. 107(3), 573–579 (2017)
Wu, D., Yau, S.T.: Negative holomorphic curvature and positive canonical bundle. Invent. Math. 204, 595–604 (2016)
Wu, D., Yau, S.T.: A remark on our paper “Negative holomorphic curvature and positive canonical bundle’’. Commun. Anal. Geom. 24, 901–912 (2016)
Yang, X.K.: RC positivity, rational connectedness and Yau’s conjecture. Camb. J. Math. 6, 183–212 (2018)
Yang, X.K.: RC-positive metrics on rationally connected manifolds. Forum Math. Sigma, 8 Paper No. e53, 19pp (2020)
Yang, X.K., Zheng, F.Y.: On real bisectional curvature for Hermitian manifolds. Trans. Am. Math. Soc. 371, 2703–2718 (2019)
Yau, S.T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100(1), 197–203 (1978)
Yau, S.T.: Problem Section, Seminar on Differential Geometry, vol. 102. Princeton Unversity Press, Princeton (1982)
Zhang, Y.S.: Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality. Math. Z. 298, 953–974 (2021)
Zhang, Y.S., Zheng, T.: On almost quasi-negative holomorphic sectional curvature. arXiv:2010.01314 (2020)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Foundation item: Supported by National Natural Science Foundation of China (Grant No. 12001490)
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Tang, K. On Almost Nonpositive k-Ricci Curvature. J Geom Anal 32, 306 (2022). https://doi.org/10.1007/s12220-022-01055-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-022-01055-2