Annals of Global Analysis and Geometry

, Volume 15, Issue 3, pp 235–242

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

  • Andrei Moroianu

DOI: 10.1023/A:1006543304443

Cite this article as:
Moroianu, A. Annals of Global Analysis and Geometry (1997) 15: 235. doi:10.1023/A:1006543304443


In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold \(\left( {M^{2m} ,g} \right)\) of even complex dimension satisfies the inequality \(\left( {M^{2m} ,g} \right)\), where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T2×N, where T2 is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.

Dirac operator Kirchberg's inequality 

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Andrei Moroianu
    • 1
  1. 1.Centre de Mathématiques de l'Ecole Polytechnique, URA 169 du CNRSPalaiseau CedexFrance