Skip to main content
SpringerLink
Log in

A Splitting Theorem for Extremal Kähler Metrics

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

Abstract

Based on recent work of S.K. Donaldson (J. Differ. Geom. 59:479–522, 2001; Q. J. Math. 56:345–356, 2005) and T. Mabuchi (Osaka J. Math. 41:563–582, 2004; Invent. Math. 159:225–243, 2005; Osaka J. Math. 46:115–139, 2009), we prove that any extremal Kähler metric in the sense of E. Calabi (in Seminar on Differential Geometry, pp. 259–290. Princeton Univ. Press, Princeton, 1982), defined on the product of polarized compact complex projective manifolds is the product of extremal Kähler metrics on each factor, provided that either the polarized manifold is asymptotically Chow semi-stable or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau (Commun. Anal. Geom. 1:473–486, 1993) about the splitting of a Kähler–Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kähler metrics.

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. \(\widetilde{\mathrm {Aut}}_{0} (X)\) is the unique connected linear algebraic subgroup of Aut0(X) such that the quotient \(\operatorname{Aut} _{0} (X)/\widetilde{{\mathrm {Aut}}}_{0} (X)\) is the Albanese torus of X [16]; its Lie algebra is the space of (real) holomorphic vector fields whose zero-set is non-empty [16, 21, 24, 25].

  2. Recall that \(\operatorname{Aut}(X,L)\) consists of the automorphisms of X which come from automorphisms of L. It is well known (see, e.g., [21, 22]) that \(\operatorname{Aut}_{0}(X,L)= \widetilde{\operatorname{Aut}}_{0}(X)\).

  3. A. Futaki [17] showed that the extremal Kähler metrics in the Kähler class of an asymptotically Chow semi-stable polarization must be CSC.

References

  1. Apostolov, V., Calderbank, D.M.J., Gauduchon, P., Tønnesen-Friedman, C.: Hamiltonian 2-forms in Kähler geometry, III. Extremal metrics and stability. Invent. Math. 173, 547–601 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bando, S., Mabuchi, T.: Uniqueness of Einstein Kähler metrics modulo connected group actions. In: Algebraic Geometry, Sendai, 1985. Adv. Stud. Pure Math., vol. 10, pp. 11–40. North-Holland, Amsterdam (1987)

    Google Scholar 

  3. Bourguignon, J.-P., Li, P., Yau, S.-T.: Upper bound for the first eigenvalue of algebraic submanifolds. Comment. Math. Helv. 69, 199–207 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Calabi, E.: Extremal Kähler metrics, seminar on differential geometry, pp. 259–290. Princeton Univ. Press, Princeton (1982)

    Google Scholar 

  5. Calabi, E.: Extremal Kähler metrics, II. In: Chavel, I., Farkas, H.M. (eds.) Differential Geometry and Complex Analysis. Springer, Berlin (1985)

    Google Scholar 

  6. Catlin, D.: The Bergman kernel and a theorem of Tian. In: Analysis and Geometry in Several Complex Variables, Katata, 1997. Trends Math., pp. 1–23. Birkhäuser, Boston (1999)

    Chapter  Google Scholar 

  7. Chen, X.X.: Space of Kähler metrics (IV)–on the lower bound of the K-energy. arXiv:0809.4081

  8. Chen, X.X., He, W.Y.: On the Calabi flow. Am. J. Math. 130, 539–570 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, X.X., Tian, G.: Geometry of Kähler metrics and holomorphic foliation by discs. Publ. Math. IHÉS 107, 1–107 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Donaldson, S.K.: Symmetric spaces, Kähler geometry and Hamiltonian dynamics. In: Eliashberg, Y., et al. (eds.) Northern California Symplectic Geometry Seminar, pp. 13–33. Amer. Math. Soc., Providence (1999)

    Google Scholar 

  11. Donaldson, S.K.: Scalar curvature and projective embeddings, I. J. Differ. Geom. 59, 479–522 (2001)

    MATH  MathSciNet  Google Scholar 

  12. Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differ. Geom. 62, 289–349 (2002)

    MATH  MathSciNet  Google Scholar 

  13. Donaldson, S.K.: Conjectures in Kähler geometry. In: Strings and Geometry. Clay Math. Proc., vol. 3, pp. 71–78. AMS, Providence (2004)

    Google Scholar 

  14. Donaldson, S.K.: Scalar curvature and projective embeddings. II. Q. J. Math. 56, 345–356 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Donaldson, S.K.: Lower bounds on the Calabi functional. J. Differ. Geom. 70, 453–472 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Fujiki, A.: On automorphism groups of compact Kähler manifolds. Invent. Math. 44, 225–258 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  17. Futaki, A.: Asymptotic Chow semi-stability and integral invariants. Int. J. Math. 15, 967–979 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  19. Huang, H., Zheng, K.: The stability of the Calabi flow near an extremal metric. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 11, 167–175 (2012)

    MATH  MathSciNet  Google Scholar 

  20. Gauduchon, P.: Fibrés hermitiens à endomorphisme de Ricci non-négatif. Bull. Soc. Math. Fr. 105, 113–140 (1977)

    MATH  MathSciNet  Google Scholar 

  21. Gauduchon, P.: Calabi’s extremal metrics: an elementary introduction. In preparation

  22. Kobayashi, S.: Transformation Groups in Differential Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 70. Springer, New York (1972)

    Book  MATH  Google Scholar 

  23. Kobayashi, S.: First Chern class and holomorphic tensor fields. Nagoya Math. J. 77, 5–11 (1980)

    MATH  MathSciNet  Google Scholar 

  24. Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Publications of the Mathematical Society of Japan, vol. 15. Princeton University Press/Iwanami Shoten, Princeton/Tokyo (1987). Memorial Lectures 5

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Luo, H.: Geometric criterion for Gieseker–Mumford stability of polarized manifolds. J. Differ. Geom. 49, 577–599 (1998)

    MATH  Google Scholar 

  27. Mabuchi, T.: Stability of extremal Kähler manifolds. Osaka J. Math. 41, 563–582 (2004)

    MATH  MathSciNet  Google Scholar 

  28. Mabuchi, T.: An energy-theoretic approach to the Hitchin–Kobayashi correspondence for manifolds. I. Invent. Math. 159, 225–243 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Mabuchi, T.: An energy-theoretic approach to the Hitchin–Kobayashi correspondence for manifolds. II. Osaka J. Math. 46, 115–139 (2009)

    MATH  MathSciNet  Google Scholar 

  30. Mabuchi, T.: Uniqueness of extremal Kähler metrics for an integral Kähler class. Int. J. Math. 15, 531–546 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mabuchi, T.: K-stability of constant scalar curvature polarization. Preprint (2008). arXiv:0812.4093

  32. Mabuchi, T.: A stronger concept of K-stability. Preprint (2009). arXiv:0910.4617

  33. Mabuchi, T.: Asymptotics of polybalanced metrics under relative stability constraints. Osaka J. Math. 48, 845–856 (2011)

    MATH  MathSciNet  Google Scholar 

  34. Ono, H., Sato, Y., Yotusani, N.: An example of asymptotic Chow unstable manifold with constant scalar curvature. Ann. Inst. Fourier 62(4), 1265–1287 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  35. Phong, D., Sturm, J.: Scalar curvature, moment maps, and the Deligne pairing. Am. J. Math. 126, 693–712 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ruan, W.: Canonical coordinates and Bergman metrics. Commun. Anal. Geom. 6, 589–631 (1998)

    MATH  Google Scholar 

  37. Stoppa, J.: K-stability of constant scalar curvature Kähler manifolds. Adv. Math. 221(4), 1397–1408 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  38. Stoppa, J., Székelyhidi, G.: Relative K-stability of extremal metrics. J. Eur. Math. Soc. 13(4), 899–909 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. Székelyhidi, G.: Extremal metrics and K-stability. Ph.D. thesis. arXiv:math/0611002

  40. Székelyhidi, G.: Extremal metrics and K-stability. Bull. Lond. Math. Soc. 39, 1–17 (2007)

    MathSciNet  Google Scholar 

  41. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990)

    MATH  Google Scholar 

  42. Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  43. Tosatti, V., Weinkove, B.: The Calabi flow with small initial energy. Math. Res. Lett. 14, 1033–1039 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  44. Wang, X.: Moment maps, Futaki invariant and stability of projective manifolds. Commun. Anal. Geom. 12, 1009–1037 (2004)

    Article  MATH  Google Scholar 

  45. Yau, S.-Y.: Nonlinear analysis in geometry. In: Monographies de L’enseignement Mathématique, no. 33, Geneva, 1986. Série des Conférences de L’union Mathématique Internationale, vol. 8 (1986)

    Google Scholar 

  46. Yau, S.-Y.: Open problems in Geometry. In: Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, CA, 1990. Proc. Sympos. Pure Math., vol. 54, pp. 1–28. AMS, Providence (1990)

    Google Scholar 

  47. Yau, S.-T.: A splitting theorem and an algebraic geometric characterization of locally hermitian symmetric spaces. Commun. Anal. Geom. 1, 473–486 (1993)

    MATH  Google Scholar 

  48. Zelditch, S.: Szegő kernel and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)

    Article  MathSciNet  Google Scholar 

  49. Zhang, S.: Heights and reductions of semi-stable varieties. Compos. Math. 104, 77–105 (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques, UQAM, Montreal, Quebec, H3C 3P8, Canada

    Vestislav Apostolov

  2. CMLS, Ecole Polytechnique, Palaiseau, 91128, France

    Hongnian Huang

Authors
  1. Vestislav Apostolov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hongnian Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongnian Huang.

Additional information

Communicated by Kang-Tae Kim.

The authors would like to thank D. Phong for valuable suggestions, as well as X.X. Chen for his interest in this work. A special acknowledgment is due to G. Székelyhidi for suggesting to us to consider approximations with balanced metrics, and for sharing with us his expertise. He decisively contributed to this project by pointing out to us the notion of Chow stability relative to a maximal torus discussed in the paper, as well as the uniqueness result in Lemma 2. The authors are grateful to the referee for a careful reading of the manuscript and suggesting a number of improvements.

The first named author was supported by an NSERC Discovery Grant and the second named author by the Fondation mathématique Jacques Hadamard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Apostolov, V., Huang, H. A Splitting Theorem for Extremal Kähler Metrics. J Geom Anal 25, 149–170 (2015). https://doi.org/10.1007/s12220-013-9417-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-013-9417-6

Keywords

Mathematics Subject Classification

Search

Navigation