Skip to main content
SpringerLink
Log in

On Gauduchon Kähler-Like Manifolds

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In a paper by Angella, Otal, Ugarte, and Villacampa, the authors conjectured that on a compact Hermitian manifold, if a Gauduchon connection other than Chern or Strominger is Kähler-like, then the Hermitian metric must be Kähler. They also conjectured that if two Gauduchon connections are both Kähler-like, then the metric must be Kähler. In this paper, we discuss some partial answers to the first conjecture, and give a proof to the second conjecture. In the process, we discovered an interesting “duality” phenomenon amongst Gauduchon connections, which seems to be intimately tied to the question, though we do not know if there is any underlying reason for that from physics.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexandrov, B., Ivanov, S.: Vanishing theorems on Hermitian manifolds. Diff. Geom. Appl. 14, 251–265 (2001)

    Article  MathSciNet  Google Scholar 

  2. Angella, D., Otal, A., Ugarte, L., Villacampa, R.: On Gauduchon connections with Kähler-like curvature. arXiv:1809.02632v2

  3. Angella, D., Ugarte, L.: Locally conformal Hermitian metrics on complex non-Kähler manifolds. Mediterr. J. Math. 13, 2105–2145 (2016)

    Article  MathSciNet  Google Scholar 

  4. Belgun, F.: On the metric structure of non-Kähler complex surfaces. Math. Ann. 317, 1–40 (2000)

    Article  MathSciNet  Google Scholar 

  5. Belgun, F.: On the metric structure of some non-Kähler complex threefolds. arXiv: 1208.4021

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

    Article  MathSciNet  Google Scholar 

  7. Enrietti, N., Fino, A., Vezzoni, L.: Tamed symplectic forms and strong Kähler with torsion metrics. J. Symplectic Geom. 10(2), 203–223 (2012)

    Article  MathSciNet  Google Scholar 

  8. Fino, A., Tardini, N.: Some remarks on Hermitian manifolds satisfying Kähler-like conditions. arXiv:2003.06582v1

  9. Fino, A., Tomassini, A.: A survey on strong KT structures. Bull. Math. Soc. Sci. Math. Roumanie 52(100), 99–116 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Fino, A., Vezzoni, L.: On the existence of balanced and SKT metrics on nilmanifolds. Proc. Am. Math. Soc. 144(6), 2455–2459 (2016)

    Article  MathSciNet  Google Scholar 

  11. Fu, J-X.: On non-Kähler Calabi-Yau threefolds with balanced metrics. In: Proceedings of the International Congress of Mathematicians. Volume II, 705-716, Hindustan Book Agency, New Delhi (2010)

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

    Article  Google Scholar 

  13. Fu, J.-X., Li, J., Yau, S.-T.: Constructing balanced metrics on some families of non-Kähler Calabi–Yau threefolds. J. Differ. Geom. 90(1), 81–129 (2012)

    Article  Google Scholar 

  14. Fu, J-X., Zhou, X.: Scalar curvatures in almost Hermitian geometry and some applications. arXiv: 1901.10130

  15. Gates, S.J., Hull, C.M., Roc̆ek, M.: Twisted multiplets and new supersymmetric nonlinear sigma models. Nuc. Phys. B 28, 157–186 (1984)

    Article  Google Scholar 

  16. Gauduchon, P.: La \(1\)-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267(4), 495–518 (1984)

    Article  MathSciNet  Google Scholar 

  17. Gauduchon, P.: Hermitian connnections and Dirac operators. Boll. Un. Mat. It 11, 257–288 (1997)

    MATH  Google Scholar 

  18. Gray, A.: Curvature identities for Hermitian and almost Hermitian manifolds. Tohoku Math. J. 28(4), 601–612 (1976)

    Article  MathSciNet  Google Scholar 

  19. Ivanov, S., Papadopoulos, G.: Vanishing theorems and string backgrounds. Classical Quantum Gravity 18, 1089–1110 (2001)

    Article  MathSciNet  Google Scholar 

  20. Khan, G., Yang, B., Zheng, F.: The set of all orthogonal complex structures on the flat \(6\)-torus. Adv. Math. 319, 451–471 (2017)

    Article  MathSciNet  Google Scholar 

  21. Li, J., Yau, S.-T.: The existence of supersymmetric string theory with torsion. J. Differ. Geom. 70(1), 143–181 (2005)

    Article  MathSciNet  Google Scholar 

  22. Liu, K.F.: Geometry of Hermitian manifolds. Internat. J. Math. 23(6), 40 (2012)

    Article  MathSciNet  Google Scholar 

  23. Liu, K.-F., Yang, X.-K.: Ricci cuvratures on Hermitian manifolds. Trans. Am. Math. Soc. 369(7), 5157–5196 (2017)

    Article  Google Scholar 

  24. Liu, K.-F., Yang, X.-K.: Hermitian harmonic maps and non-degenerate curvatures. Math. Res. Lett. 21(4), 831–862 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Otal, A., Ugarte, L., Villacampa, R.: Invariant solutions to the Strominger system and the heterotic equations of motion. arXiv:1604.02851v3

  27. Popovici, D.: Limits of projective manifolds under holomorphic deformations: Hodge numbers and strongly Gauduchon metrics. Invent. Math. 194(3), 515–534 (2013)

    Article  MathSciNet  Google Scholar 

  28. Streets, J.: Pluriclosed flow and the geometrization of complex surfaces. Geometric Anal. Progress Math. 333, 471–510 (2020)

    Article  MathSciNet  Google Scholar 

  29. Strominger, A.: Superstrings with Torsion. Nuclear Phys. B 274, 253–284 (1986)

    Article  MathSciNet  Google Scholar 

  30. Székelyhidi, G., Tosatti, V., Weinkove, B.: Gauduchon metrics with prescribed volume form. Acta Math. 219(1), 181–211 (2017)

    Article  MathSciNet  Google Scholar 

  31. Tosatti, V.: Non-Kähler Calabi-Yau manifolds. In: Analysis, complex geometry, and mathematical physics: in honor of Duong H. Phong, pp. 261–277 Contemporary Mathematics, 644, American Mathematical Society, Providence, RI, 2015. arXiv: 1401.4797

  32. Tseng, L.-S., Yau, S.-T.: Non-Kähler Calabi-Yau manifolds. String-Math 2011. In: Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI 85, pp. 241–254 (2012)

  33. Vezzoni, L., Yang, B., Zheng, F.: Lie groups with flat Gauduchon connections. Math. Zeit. 293(1–2), 597–608 (2019)

    Article  MathSciNet  Google Scholar 

  34. Wang, Q., Yang, B., Zheng, F.: On Bismut flat manifolds. Trans. Am. Math. Soc. 373, 5747–5772 (2020)

    Article  MathSciNet  Google Scholar 

  35. Yang, B., Zheng, F.: On curvature tensors of Hermitian manifolds. Commun. Anal. Geom. 26(5), 1193–1220 (2018)

    Article  MathSciNet  Google Scholar 

  36. Yang, B., Zheng, F.: On compact Hermitian manifolds with flat Gauduchon conmnections. Acta Math. Sin. (English Series) 34, 1259–1268 (2018)

    Article  MathSciNet  Google Scholar 

  37. Yau, S.-T., Zhao, Q., Zheng, F.: On Strominger Kähler-like manifolds with degenerate torsion. arXiv:1908.05322v2

  38. Zhao, Q., Zheng, F.: Strominger connection and pluriclosed metrics. arXiv:1904.06604v3

  39. Zhao, Q., Zheng, F.: Complex nilmanifolds and Kähler-like connections. J. Geom. Phys. 146, 14–49 (2019)

    Article  Google Scholar 

  40. Zheng, F.: Some recent progress in non-Kähler geometry. Sci. China Math. 62(11), 2423–2434 (2019)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank mathematicians Jixiang Fu, Gabriel Khan, Kefeng Liu, Luigi Vezzoni, Bo Yang, Xiaokui Yang, Shing-Tung Yau, and Xianchao Zhou for their interests and/or help. We are also indebted to the work [2, 14, 26] which inspired this study.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Quanting Zhao

  2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Fangyang Zheng

Authors
  1. Quanting Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangyang Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

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

Zhao is partially supported by National Natural Science Foundations of China with the Grant No. 11801205. Zheng is partially supported by National Natural Science Foundations of China with the Grant Nos. 12071050, 12141101, and by a Chongqing Grant cstc2021ycjh-bgzxm0139.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhao, Q., Zheng, F. On Gauduchon Kähler-Like Manifolds. J Geom Anal 32, 110 (2022). https://doi.org/10.1007/s12220-022-00868-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-00868-5

Keywords

Mathematics Subject Classification

Search

Navigation