Proceedings of the Japan Academy, Series A, Mathematical Sciences

An obstruction to semistability of manifolds

Toshiki Mabuchi and Yasuhiro Nakagawa

Full-text not supplied by publisher


This is an announcement of our result showing that the Bando-Calabi-Futaki character for an integral Kähler class, if any, of a compact complex manifold is regarded as an obstruction to semistability of the manifold. This develops a pioneering work of Fujiki [3].

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 4 (2001), 47-49.

First available in Project Euclid: 23 May 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Secondary: 14L24: Geometric invariant theory [See also 13A50] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

The Bando-Calabi-Futaki character obstruction semistability Tian Chow points


Mabuchi, Toshiki; Nakagawa, Yasuhiro. An obstruction to semistability of manifolds. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 4, 47--49. doi:10.3792/pjaa.77.47.

Export citation


  • Bando, S.: An obstruction for Chern class forms to be harmonic (preprint).
  • Calabi, E.: Extremal Kähler metrics II. Differential Geometry and Complex Analysis (eds. I. Chavel, I., and Farkas, H. M.). Springer, Heidelberg, pp. 95–114 (1985).
  • Fujiki, A.: Moduli space of polarized algebraic manifolds and Kähler metrics. Sugaku, 42, 231–243 (1990); Sugaku Expositions, 5, 173–191 (1992) (English transl.).
  • Futaki, A.: An obstruction to the existence of Kähler Einstein metrics. Invent. Math., 73, 437–443 (1983).
  • Futaki, A.: On compact Kähler manifolds of constant scalar curvature. Proc. Japan Acad., 59A, 401–402 (1983).
  • Mabuchi, T., and Nakagawa, H.: The Bando-Calabi-Futaki character as an obstruction to semistability (preprint).
  • Mumford, D.: Stability of porjective varieties. Enseignement Math., 23, 39–110 (1977).
  • Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math., 130, 1–37 (1997).