Science China Mathematics

, Volume 62, Issue 11, pp 2423–2434 | Cite as

Some recent progress in non-Kähler geometry

  • Fangyang ZhengEmail author


In this paper, we discuss some recent progress in the study of non-Kähler manifolds, in particular the Hermitian geometry of flat canonical connections and Kähler-like connections. We also discuss a number of conjectures and open questions in this direction.


non-Kähler manifolds Hermitian geometry Kähler-like metrics 


53026 53C55 



The author thanks his former coauthors Gabriel Khan, Luigi Vezzoni, Qingsong Wang, Bo Yang, and Quanting Zhao for their collaborations. The survey is based on the joint works with them.


  1. 1.
    Alexandrov B, Ivanov S. Vanishing theorems on Hermitian manifolds. Differential Geom Appl, 2001, 14: 251–265MathSciNetCrossRefGoogle Scholar
  2. 2.
    Angella D, Otal A, Ugarte L, et al. On Gauduchon connections with Kähler-like curvature. ArXiv:1809.02632, 2018Google Scholar
  3. 3.
    Atiyah M, Hitchin N, Singer I-M. Self-duality in four-dimensional Riemannian geometry. Proc R Soc Lond Ser A Math Phys Eng Sci, 1978, 362: 425–461MathSciNetCrossRefGoogle Scholar
  4. 4.
    Belgun F. On the metric structure of non-Kähler complex surfaces. Math Ann, 2000, 317: 1–40MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bismut J-M. A local index theorem for non-Kähler manifolds. Math Ann, 1989, 284: 681–699MathSciNetCrossRefGoogle Scholar
  6. 6.
    Boothby W. Hermitian manifolds with zero curvature. Michigan Math J, 1958, 5: 229–233MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borisov L, Salamon S, Viaclovsky J. Twistor geometry and warped product orthogonal complex structures. Duke Math J, 2011, 156: 125–166MathSciNetCrossRefGoogle Scholar
  8. 8.
    Calabi E, Eckmann A. A class of compact, complex manifolds which are not algebraic. Ann of Math (2), 1953, 58: 494–500MathSciNetCrossRefGoogle Scholar
  9. 9.
    Deligne P, Griffiths P, Morgan J, et al. Real homotopy theory of Kähler manifolds. Invent Math 1975, 29: 245–274MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fu J-X. On non-Kähler Calabi-Yau threefolds with balanced metrics. In: Proceedings of the International Congress of Mathematicians, vol. 2. New Delhi: Hindustan Book Agency, 2010, 705–716Google Scholar
  11. 11.
    Fu J-X, Wang Z-Z, Wu D. Form-type Calabi-Yau equations. Math Res Lett, 2010, 17: 887–903MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fu J-X, Wang Z-Z, Wu D. Semilinear equations, the γk function, and generalized Gauduchon metrics. J Eur Math Soc (JEMS), 2013, 15: 659–680MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fu J-X, Yau S-T. The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampere equation. J Differential Geom, 2008, 78: 369–428MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gauduchon P. Hermitian connections and Dirac operators. Boll Unione Mat Ital (9), 1997, 11: 257–288MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gray A. Curvature identities for Hermitian and almost Hermitian manifolds. Tohoku Math J (2), 1976, 28: 601–612MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hitchin N. Käahlerian twistor spaces. Proc Lond Math Soc (3), 1981, 43: 133–150CrossRefGoogle Scholar
  17. 17.
    Khan G, Yang B, Zheng F. The set of all orthogonal complex strutures on the flat 6-torus. Adv Math, 2017, 319: 451–471MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li J, Yau S-T. The existence of supersymmetric string theory with torsion. J Differential Geom, 2005, 70: 143–181MathSciNetCrossRefGoogle Scholar
  19. 19.
    Liu K-F, Yang X-K. Geometry of Hermitian manifolds. Internat J Math, 2012, 23: 1250055MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu K-F, Yang X-K. Ricci cuvratures on Hermitian manifolds. ArXiv: 1404.2481, 2014Google Scholar
  21. 21.
    Liu K-F, Yang X-K. Hermitian harmonic maps and non-degenerate curvatures. Math Res Lett, 2014, 21: 831–862MathSciNetCrossRefGoogle Scholar
  22. 22.
    Milnor J. Curvatures of left invariant metrics on Lie groups. Adv Math, 1976, 21: 293–329MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pittie H. The Dolbeault-cohomology ring of a compact, even-dimensional Lie group. Proc Indian Acad Sci Math Sci, 1988, 98: 117–152MathSciNetCrossRefGoogle Scholar
  24. 24.
    Salamon S. Orthogonal complex structures. In: Differential geometry and applications. Burno: Masaryk University, 1996, 103–117Google Scholar
  25. 25.
    Salamon S, Viaclovsky J. Orthogonal complex structures on domains in ℝ4. Math Ann, 2009, 343: 853–899MathSciNetCrossRefGoogle Scholar
  26. 26.
    Samelson H. A class of complex analytic manifolds. Port Math, 1953, 12: 129–132MathSciNetzbMATHGoogle Scholar
  27. 27.
    Streets J, Tian G. A parabolic flow of pluriclosed metrics. Int Math Res Not IMRN, 2010, 16: 3101–3133MathSciNetzbMATHGoogle Scholar
  28. 28.
    Streets J, Tian G. Regularity results for pluriclosed flow. Geom Topol, 2013, 17: 2389–2429MathSciNetCrossRefGoogle Scholar
  29. 29.
    Strominger A. Superstrings with Torsion. Nuclear Phys B, 1986, 274: 253–284MathSciNetCrossRefGoogle Scholar
  30. 30.
    Szekelyhidi G, Tosatti V, Weinkove B. Gauduchon metrics with prescribed volume form. Acta Math, 2017, 219: 181–211MathSciNetCrossRefGoogle Scholar
  31. 31.
    Taubes C. The existence of anti-self-dual conformal structures. J Differential Geom, 1992, 36: 163–253MathSciNetCrossRefGoogle Scholar
  32. 32.
    Tosatti V. Non-Käahler Calabi-Yau manifolds. In: Analysis, Complex Geometry, and Mathematical Physics. Contemporary Mathematics, vol. 644. Providence: Amer Math Soc, 2015, 261–277CrossRefGoogle Scholar
  33. 33.
    Tosatti V, Weinkove B. The complex Monge-Ampére equation on compact Hermitian manifolds. J Amer Math Soc 2010, 23: 1187–1195MathSciNetCrossRefGoogle Scholar
  34. 34.
    Tseng L-S, Yau S-T. Non-Kähler Calabi-Yau manifolds. In: Proceedings of Symposia in Pure Mathematics, vol. 85. Providence: Amer Math Soc, 2012, 241–254Google Scholar
  35. 35.
    Vezzoni L, Yang B, Zheng F. Lie groups with flat Gauduchon connections. Math Z, 2019, in pressGoogle Scholar
  36. 36.
    Wang Q, Yang B, Zheng F. On Bismut flat manifolds. ArXiv:1603.07058, 2016Google Scholar
  37. 37.
    Yang B, Zheng F. On curvature tensors of Hermitian manifolds. Comm Anal Geom, 2018, 26: 1193–1220MathSciNetCrossRefGoogle Scholar
  38. 38.
    Yang B, Zheng F. On compact Hermitian manifolds with flat Gauduchon conmnections. Acta Math Sin Engl Ser, 2018, 34: 1259–1268MathSciNetCrossRefGoogle Scholar
  39. 39.
    Zhao Q, Zheng F. Strominger connections and pluriclosed metrics. ArXiv:1904.06604, 2019Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA
  2. 2.Department of MathematicsZhejiang Normal UniversityJinhuaChina

Personalised recommendations