Skip to main content
SpringerLink
Log in

Conformally Einstein–Maxwell Kähler metrics and structure of the automorphism group

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let (M, g) be a compact Kähler manifold and f a positive smooth function such that its Hamiltonian vector field \(K = J\mathrm {grad}_g f\) for the Kähler form \(\omega _g\) is a holomorphic Killing vector field. We say that the pair (g, f) is conformally Einstein–Maxwell Kähler metric if the conformal metric \(\tilde{g} = f^{-2}g\) has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein–Maxwell Kähler manifolds, extending the Lichnerowicz–Matsushima Theorem for constant scalar curvature Kähler manifolds. More generally we consider extensions of Calabi functional and extremal Kähler metrics, and prove an extension of Calabi’s theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.

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

Notes

  1. A.Lahdili obtained independently a proof of Theorem 2.1 in [19]

References

  1. Apostolov, V., Calderbank, D.M.J., Gauduchon, P.: Ambitoric geometry I: Einstein metrics and extremal ambikähler structures. J. Reine Angew. Math. 721, 109–147 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Apostolov, V., Calderbank, D .M .J., Gauduchon, P.: Ambitoric geometry II: extremal toric surfaces and Einstein 4-orbifolds. Ann. Sci. Éc. Norm. Supér. (4) 48(5), 1075–1112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apostolov, V., Maschler, G.: Conformally Kähler, Einstein–Maxwell geometry, preprint, arXiv:1512.06391

  4. Bérard-Bergery, L.: Lionel . Sur de nouvelles variétés riemanniennes d’Einstein. (French) [Some new Einstein Riemannian manifolds] Institut Élie Cartan, 6, 1–60, Inst. Élie Cartan, 6, Univ. Nancy, Nancy (1982)

  5. Besse, A.: Einstein manifolds. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  6. Calabi, E.: Extremal Kähler metrics II, Differential geometry and complex analysis. In: Chavel, I., Farkas, H.M. (eds.), 95–114, Springer, Berlin (1985)

  7. Carrell, J.B., Lieberman, D.I.: Vector fields and Chern Numbers. Math. Annalen 225, 263–273 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X.-X., Lebrun, C., Weber, B.: On conformally Kähler, Einstein manifolds. J. Am. Math. Soc. 21(4), 1137–1168 (2008)

    Article  MATH  Google Scholar 

  9. Donaldson, S.K.: Remarks on gauge theory, complex geometry and \(4\)-manifold topology. In: Atiyah, I. (eds) Fields medallists lectures, World Scientific, 384–403 (1997)

  10. Fujiki, A.: Moduli space of polarized algebraic manifolds and Kähler metrics. Sugaku Expos. 5, 173–191 (1992)

    MATH  Google Scholar 

  11. Futaki, A.: An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73, 437–443 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  12. Futaki, A.: On compact Kähler manifolds of constant scalar curvature. Proc. Jpn. Acad. Ser. A 59, 401–402 (1983)

    Article  MATH  Google Scholar 

  13. Futaki, A.: Harmonic total Chern forms and stability. Kodai Math. J. 29(3), 346–369 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Futaki, A.: Holomorphic vector fields and perturbed extremal Kähler metrics. J. Symplectic Geom. 6(2), 127–138 (2008). arXiv:math.DG/0702721

    Article  MathSciNet  MATH  Google Scholar 

  15. Futaki, A., Ono, H.: Volume minimization and conformally Kähler, Einstein-Maxwell geometry. J. Math. Soc. Japan. arXiv:1706.07953 (to appear)

  16. Futaki, A., Ono, H., Wang, G.: Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differ. Geom. 83, 585–635 (2009)

    Article  MATH  Google Scholar 

  17. Gauduchon, P.: Calabis extremal metrics: an elementary introduction, Lecture Notes

  18. Koca, C., Tønnesen-Friedman, C.W.: Strongly Hermitian Einstein- Maxwell solutions on ruled surfaces. Ann. Glob. Annl. Geom. 50, 29–46 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lahdili, A.: Automorphisms and deformations of conformally Kähler, Einstein–Maxwell metrics, arXiv:1708.01507

  20. LeBrun, C.: The Einstein-Maxwell equations, Kähler metrics, and Hermitian geometry. J. Geom. Phys. 91, 163–171 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. LeBrun, C.: The Einstein-Maxwell equations and conformally Kähler geometry. Commun. Math. Phys. 344, 621–653 (2016)

    Article  MATH  Google Scholar 

  22. LeBrun, C., Simanca, R.S.: Extremal Kähler metrics and complex deformation theory. Geom. Func. Anal. 4, 298–336 (1994)

    Article  MATH  Google Scholar 

  23. Lichnerowicz, A.: Géometrie des groupes de transformations, Dunod, Paris (1958)

  24. Matsushima, Y.: Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kaehlérienne. Nagoya Math. J. 11, 145–150 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  25. Matsushima, Y.: Holomorphic vector fields on compact Kähler manifolds, Conference Board of the Mathematical Sciences, Regional Conference Series in Mathematics, No. 7. American Mathematical Society, Providence, RI (1971)

  26. Martelli, D., Sparks, J., Yau, S.-T.: Sasaki–Einstein manifolds and volume minimisation. Commun. Math. Phys. 280(3), 611–673 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Page, D.: A compact rotating gravitational instanton. Phys. Lett. B 79, 235–238 (1978)

    Article  Google Scholar 

  28. Tian, G., Zhu, X.-H.: A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. Comment. Math. Helv. 77, 297–325 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, L.-J.: Hessians of the Calabi functional and the norm function. Ann. Global Anal. Geom 29(2), 187–196 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Author notes

  1. Akito Futaki

    Present address: Yau Mathematical Sciences Center, Tsinghua University, Hai Dian District, Beijing, 100084, China

Authors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan

    Akito Futaki

  2. Department of Mathematics, Saitama University, 255 Shimo-Okubo, Sakura-Ku, Saitama, 380-8570, Japan

    Hajime Ono

Authors
  1. Akito Futaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hajime Ono
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Akito Futaki.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Futaki, A., Ono, H. Conformally Einstein–Maxwell Kähler metrics and structure of the automorphism group. Math. Z. 292, 571–589 (2019). https://doi.org/10.1007/s00209-018-2112-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-018-2112-3

Search

Navigation