Homogeneous almost-Kähler manifolds and the Chern–Einstein equation


Given a non-compact semisimple Lie group G we describe all homogeneous spaces G / L carrying an invariant almost-Kähler structure \((\omega ,J)\). When L is abelian and G is of classical type, we classify all such spaces which are Chern–Einstein, i.e. which satisfy \(\rho = \lambda \omega \) for some \(\lambda \in {\mathbb {R}}\), where \(\rho \) is the Ricci form associated to the Chern connection.

This is a preview of subscription content, access via your institution.


  1. 1.

    Apostolov, V., Drăghici, T.: The curvature and the integrability of almost-Kähler manifolds: a survey. Symplectic Contact Topol. 35, 25–53 (2003)

    MATH  Google Scholar 

  2. 2.

    Alekseevsky, D.V., Perelomov, A.M.: Invariant Kähler-Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl. 20, 171–182 (1986)

    Article  Google Scholar 

  3. 3.

    Borel, A.: Compact Clifford-Klein forms of symmetric spaces. Topology 2, 111–122 (1963)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bordermann, M., Forger, M., Römer, H.: Homogeneous Kähler Manifolds: paving the way towards new supersymmetric Sigma Models. Comm. Math. Phys. 102, 605–647 (1986)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Donaldson, S.: Remarks on Gauge theory, complex geometry and \(4\)-manifolds topology. In: Atiyah, M., Iagolnitzer, D. (eds.) The fields medallists lectures, pp. 384–403. World Scientific, Singapore (1997)

    Google Scholar 

  6. 6.

    Davidov, J., Grantcharov, G., Muskarov, O.: Curvature properties of the Chern connection of twistor spaces. Rocky Mountain J. Math. 39, 27–48 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Della Vedova, A.: Special homogeneous almost complex structures on symplectic manifolds. J. Sympl. Geom. 17, 1251–1295 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Della Vedova, A., Gatti, A.: Almost Kähler geometry of adjoint orbits of semisimple Lie groups (2018). arXiv:1811.06958v2 [mathSG]

  9. 9.

    Fine, J., Panov, D.: Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle. Geom. Topol. 14, 1723–1763 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gorbatsevich, V.V., Onishchik, A.L., Vinberg, E.B.: Foundations of Lie theory and Lie transformation groups. Springer Verlag, Berlin (1997)

    MATH  Google Scholar 

  11. 11.

    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, Inc, Cambridge (1978)

    MATH  Google Scholar 

  12. 12.

    Lejmi, M.: Stability under deformations of Hermite-Einstein almost Kähler metrics. Ann. Inst. Fourier (Grenoble) 64, 2251–2263 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Lejmi, M.: Extremal almost-Kähler metrics. Intern. J. Math. 21, 1639–1662 (2010)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Podestà, F.: Homogeneous Hermitian manifolds and special metrics. Transform. Groups 23, 1129–1147 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Podestà, F., Raffero, A.: Homogeneous symplectic half-flat \(6\)-manifolds. Ann. Glob. Anal. Geom. 55, 1–15 (2019)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Vernier, C.: Almost-Kähler smoothings of compact complex surfaces with \(A_1\) singularities (2018). arXiv:1806.07773v1

  17. 17.

    Wolf, J.A., Gray, A.: Homogeneous spaces defined by Lie group automorphisms. II. J. Differ. Geom. 2, 115–159 (1968)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Fabio Podestà.

Additional information

Publisher's Note

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

Dmitri V. Alekseevsky was partially supported by grant no. 18-00496S of the Czech Science Foundation.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alekseevsky, D.V., Podestà, F. Homogeneous almost-Kähler manifolds and the Chern–Einstein equation. Math. Z. 296, 831–846 (2020). https://doi.org/10.1007/s00209-019-02446-y

Download citation


  • Symplectic manifolds
  • Homogeneous spaces
  • Chern Ricci form

Mathematics Subject Classification

  • 53C25
  • 53C30