Tohoku Mathematical Journal

On the existence of Kähler metrics of constant scalar curvature

Kenji Tsuboi

Full-text: Open access


For certain compact complex Fano manifolds $M$ with reductive Lie algebras of holomorphic vector fields, we determine the analytic subvariety of the second cohomology group of $M$ consisting of Kähler classes whose Bando-Calabi-Futaki character vanishes. Then a Kähler class contains a Kähler metric of constant scalar curvature if and only if the Kähler class is contained in the analytic subvariety. On examination of the analytic subvariety, it is shown that $M$ admits infinitely many nonhomothetic Kähler classes containing Kähler metrics of constant scalar curvature but does not admit any Kähler-Einstein metric.

Article information

Tohoku Math. J. (2), Volume 61, Number 2 (2009), 241-252.

First available in Project Euclid: 24 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Kähler manifold constant scalar curvature Bando-Calabi-Futaki character


Tsuboi, Kenji. On the existence of Kähler metrics of constant scalar curvature. Tohoku Math. J. (2) 61 (2009), no. 2, 241--252. doi:10.2748/tmj/1245849446.

Export citation


  • S. Bando, An obstruction for Chern class forms to be harmonic, Kodai Math. J. 29 (2006), 337--345.
  • E. Calabi, Extremal Kähler metrics II, Differential geometry and complex analysis, (I. Chavel and H. M. Farkas eds.), 95--114, Springer-Verlag, Berline-Heidelberg-New York, 1985.
  • A. Futaki, An obstruction to the existence of Einstein-Kähler metrics, Invent. Math. 73 (1983), 437--443.
  • A. Futaki, On compact Kähler manifold of constant scalar curvature, Proc. Japan Acad. Ser. A 59 (1983), 401--402.
  • A. Futaki, Kähler-Einstein metrics and integral invariants, Lecture Notes in Math. 1314, Springer-Verlag, Berlin, 1988.
  • A. Futaki and S. Morita, Invariant polynomials of the automorphism group of a compact complex manifold, J. Differential Geom. 21 (1985), 135--142.
  • A. Futaki and K. Tsuboi, Fixed point formula for characters of automorphism groups associated with Kähler classes, Math. Res. Lett. 8 (2001), 495--507.
  • A. D. Hwang, On existence of Kähler metrics with constant scalar curvature, Osaka J. Math. 31 (1994), 561--595.
  • A. T. Huckleberry and D. M. Snow, Almost-homogeneous Kähler manifolds with hypersurface orbits, Osaka J. Math. 19 (1982), 763--786.
  • J. Kazdan and F. Warner, Prescribing curvatures, Proc. Sympos. Pure Math. 27 (1975), 309--319.
  • S. Kobayashi, Transformation groups in differential geometry, Springer-Verlag, Berlin-Heidelberg-New York, 1972.
  • A. Lichnerowicz, Sur les transformations analytiques d'une variété Kählerienne compacte, 1959, Colloque Geom. Diff. Global (Bruxelles, 1958), 11--26, Centre Belge Rech. Math., Louvain.
  • A. Lichnerowicz, Isométrie et transformations analytiques d'une variété Kählerienne compacte, Bull. Soc. Math. France 87 (1959), 427--437.
  • Y. Matsushima, Sur la structure du groupe d'homéomorphismes d'une certaine variété Kaehlérienne, Nagoya Math. J. 11 (1957), 145--150.
  • Y. Nakagawa, Bando-Calabi-Futaki character of compact toric manifolds, Tohoku Math. J. 53 (2001), 479--490.
  • Y. Nakagawa, The Bando-Calabi-Futaki character and its lifting to a group character, Math. Ann. 325 (2003), 31--53.
  • G. Tian, Kähler-Einstein metrics on algebraic manifolds, in: Proc. C.I.M.E. conference on Transcendental methods in algebraic geometry, Lecture Notes in Math. 1646, Springer-Verlag, Berlin-Heidelberg-New York, 1996, 143--185.