Abstract
A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov—Davidov—Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the special case of Strominger Kähler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the Kähler symmetries.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Angella, D., Otal, A., Ugarte, L., et al.: On Gauduchon connections with Kähler-like curvature. arXiv:1809.02632, to appear in Comm. Anal. Geom.
Apostolov, V., Davidov, J., Muskarov, O.: Compact self-dual Hermitian surfaces. Trans. Amer. Math. Soc., 348, 3051–3063 (1996)
Balas, A.: Compact Hermitian manifolds of constant holomorphic sectional curvature. Math. Z., 189, 193–210 (1985)
Balas, A., Gauduchon, P.: Any Hermitian metric of constant nonpositive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler. Math. Z., 190, 39–43 (1985)
Barberis, M. L., Dotti, I., Fino, A.: Hyper-Kähler quotients of solvable Lie groups. J. Geom. Phys., 56(4), 691–711 (2006)
Boothby, W.: Hermitian manifolds with zero curvature. Michigan Math. J., 5(2), 229–233 (1958)
Chen, H., Chen, L., Nie, X.: Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics. Sci. China Math., 64, 763–780 (2021)
Gauduchon, P: La 1-forme de torsion d’une variété hermitienne compacte. Math. Ann., 267(4), 495–518 (1984)
Gauduchon, P.: Hermitian connections and Dirac operators. Boll. Un. Mat. Ital. B (7), 11(2), 257–288 (1997)
Li, Y., Zheng, F.: Complex nilmanifolds with constant holomorphic sectional curvature. Proc. Amer. Math. Soc., 150(1), 319–326 (2022)
Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math., 21(3), 293–329 (1976)
Salamon, S.M.: Complex structures on nilpotent Lie algebras. J. Pure Appl. Algebra, 157(2–3), 311–333 (2001)
Tang, K.: Holomorphic sectional curvature and Kähler-like metric. Sci. Sin. Math. (Chin. Ser.), 50, 1–12 (2021)
Vezzoni, L., Yang, B., Zheng, F.: Lie groups with flat Gauduchon connections. Math. Z., 293, 597–608 (2019)
Wang, Q., Yang, B., Zheng, F.: On Bismut flat manifolds. Trans. Amer. Math. Soc., 373(8), 5747–5772 (2020)
Yang, B., Zheng, F.: On curvature tensors of Hermitian manifolds. Comm. Anal. Geom., 26(5), 1195–1222 (2018)
Yang, B., Zheng, F.: On compact Hermitian manifolds with a flat Gauduchon connection. Acta Math. Sin., Engl. Ser., 34(8), 1259–1268 (2018)
Yang, X., Zheng, F.: On real bisectional curvature for Hermitian manifolds. Trans. Amer. Math. Soc., 371(4), 2703–2718 (2019)
Yau, S. T., Zhao, Q., Zheng, F.: On Strominger Kähler-like manifolds with degenerate torsion. arXiv: 1908.05032
Zhao, Q., Zheng, F.: Complex nilmanifolds and Kähler-like connections. J. Geom. Phys., 146, 103512, 9 pp. (2019)
Zhao, Q., Zheng, F.: Strominger connection and pluriclosed metrics. arXiv:1904.06604
Zhou, W., Zheng, F.: Hermitian threefolds with vanishing real bisectional curvature. arXiv:2103.04296, to appear in Sci. Sin. Math. (Chin. Ser.)
Acknowledgements
The second named author would like to thank mathematicians Haojie Chen, Xiaolan Nie, Kai Tang, Bo Yang, Xiaokui Yang, and Quanting Zhao for their interests and/or helpful discussions. We would also like to take this opportunity to thank anonymous referees for a number of useful suggestion/clarification which helped to improve the readability of the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by NSFC (Grant No. 12071050) and Chongqing Normal University
Rights and permissions
About this article
Cite this article
Rao, P.P., Zheng, F.Y. Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature. Acta. Math. Sin.-English Ser. 38, 1094–1104 (2022). https://doi.org/10.1007/s10114-022-1046-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-022-1046-1