Skip to main content
Log in

A new positivity condition for the curvature of Hermitian manifolds

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript


In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kähler case. We derive a Bochner formula for closed (1, 1)-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfies our positivity condition, then any class \(\alpha \in H^{1, 1}_{BC}(X)\) can be represented by a closed (1, 1)-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Bishop, R.L., Goldberg, S.I.: On the second cohomology group of a Kähler manifold of positive curvature. Proc. Am. Math. Soc. 16, 119–122 (1965)

    MATH  Google Scholar 

  2. Bismut, J.-M.: A local index theorem for non-Kähler manifolds. Math. Ann. 284(4), 681–699 (1989)

    Article  MathSciNet  Google Scholar 

  3. Chau, A., Tam, L.F.: On quadratic orthogonal bisectional curvature. J. Differ. Geom. 92(2), 187–200 (2012)

    Article  MathSciNet  Google Scholar 

  4. Fei, T., Phong, D.H.: Unification of the Kähler–Ricci flow and anomaly flows. Surv. Differ. Geom. 23, 89–104 (2018)

    Article  Google Scholar 

  5. Fei, T., Yau, S.T.: Invariant solutions to the Strominger system on complex Lie groups and their quotients. Commun. Math. Phys. 338(3), 1183–1195 (2015)

    Article  MathSciNet  Google Scholar 

  6. Fei, T., Huang, Z., Picard, S.: A construction of infinitely many solutions to the Strominger system. arXiv:1703.10067

  7. Fu, J., Yau, S.T.: The theory of superstring with flux on non-Kähler manifolds and the complex Monge–Ampere equation. J. Differ. Geom. 78(3), 369–428 (2008)

    Article  Google Scholar 

  8. Gauduchon, P.: Le théoréme de l’excentricité nulle (French). C. R. Acad. Sci. Paris Sér. A-B 285(5), A387–390 (1977)

    MathSciNet  MATH  Google Scholar 

  9. Howard, A., Smyth, B., Wu, H.: On compact Kähler manifolds of nonnegative bisectional curvature. I. Acta Math. 147(1–2), 51–56 (1981)

    Article  MathSciNet  Google Scholar 

  10. Hull, C., Witten, E.: Supersymmetric sigma models and the heterotic string. Phys. Lett. B 160(6), 398–402 (1985)

    Article  MathSciNet  Google Scholar 

  11. Mok, N.: The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J. Differ. Geom. 27(2), 179–214 (1988)

    Article  Google Scholar 

  12. Mok, N., Siu, Y.T., Yau, S.T.: The Poincaré–Lelong equation on complete Kähler manifolds. Compos. Math. 44(1–3), 183–218 (1981)

    MATH  Google Scholar 

  13. Mori, S.: Projective manifolds with ample tangent bundles. Ann. Math. (2) 110(3), 593–606 (1979)

    Article  MathSciNet  Google Scholar 

  14. Ni, L., Tam, L.F.: Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature. J. Differ. Geom. 64(3), 457–524 (2003)

    Article  Google Scholar 

  15. Ni, L., Tam, L.F.: The Poincaré–Lelong equation via the Hodge–Laplace heat equation. Compos. Math. 149(11), 1856–1870 (2013)

    Article  MathSciNet  Google Scholar 

  16. Ornea, L., Gauduchon, P.: Locally conformally Kähler metrics on Hopf surfaces. Ann. Inst. Fourier (Grenoble) 48(4), 1107–1127 (1998)

    Article  MathSciNet  Google Scholar 

  17. Ornea, L., Verbitsky, M.: Structure theorem for compact Vaisman manifolds. Math. Res. Lett. 10(5–6), 799–805 (2003)

    Article  MathSciNet  Google Scholar 

  18. Ornea, L., Verbitsky, M.: Locally conformal Kähler manifolds with potential. Math. Ann. 348(1), 25–33 (2010)

    Article  MathSciNet  Google Scholar 

  19. Ornea, L., Verbitsky, M.: LCK rank of locally conformally Kähler manifolds with potential. J. Geom. Phys. 107, 92–98 (2016)

    Article  MathSciNet  Google Scholar 

  20. Ornea, L., Verbitsky, M.: Hopf surfaces in locally conformally Kähler manifolds with potential. Geom. Dedicata (2019)

  21. Phong, D.H.: Geometric partial differential equations from unified string theories. In: Proceedings of the ICCM 2018, Taipei (2018)

  22. Phong, D.H., Picard, S., Zhang, X.: Anomaly flows. Commun. Anal. Geom. 26(4), 955–1008 (2018)

    Article  MathSciNet  Google Scholar 

  23. Phong, D.H., Picard, S., Zhang, X.: The anomaly flow on unimodular Lie groups, Advances in Complex Geometry, Contemp. Math., vol. 735. Amer. Math. Soc., Providence, pp. 217–237 (2019)

  24. Siu, Y.T., Yau, S.T.: Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59(2), 189–204 (1980)

    Article  MathSciNet  Google Scholar 

  25. Streets, J., Tian, G.: A parabolic flow of pluriclosed metrics. Int. Math. Res. Not. IMRN 16, 3101–3133 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Streets, J., Tian, G.: Regularity results for pluriclosed flow. Geom. Topol. 17(4), 2389–2429 (2013)

    Article  MathSciNet  Google Scholar 

  27. Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274(2), 253–284 (1986)

    Article  MathSciNet  Google Scholar 

  28. Ustinovskiy, Y.: On the Structure of Hermitian Manifolds with Semipositive Griffiths Curvature. Trans. Amer. Math. Soc. (2020)

  29. Vaisman, I.: Generalized Hopf manifolds. Geom. Dedicata 13(3), 231–255 (1982)

    Article  MathSciNet  Google Scholar 

  30. Verbitsky, M.: Theorems on the vanishing of cohomology for locally conformally hyper-Kähler manifolds. (Russian) , Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., pp. 64–91 [translation in Proc. Steklov Inst. Math. 2004, no. 3(246), 54–78 (2004)]

  31. Wang, Q., Yang, B., Zheng, F.: On Bismut flat manifolds. arXiv:1603.07058

  32. Zhao, Q., Zheng, F.: Strominger connection and pluriclosed metrics. arXiv:1904.06604

Download references


I would like to thank my advisor Duong Phong for many helpful suggestions and for his constant support and encouragement. I am also grateful to Nikita Klemyatin for bringing to my attention the paper [19].

Author information

Authors and Affiliations


Corresponding author

Correspondence to Freid Tong.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported in part by NSF grant DMS-1855947.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tong, F. A new positivity condition for the curvature of Hermitian manifolds. Math. Z. 298, 1175–1185 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: