On Kirchberg's inequality for compact Kähler manifolds of even complex dimension (Q1361045)

In [Math. Nachr. 97, 117-146 (1980; Zbl 0462.53027)] \textit{Th. Friedrich} proved a first inequality for the eigenvalues of the Dirac operator on a compact Riemannian spin manifold, giving a lower bound which contains the scalar curvature of the underlying manifold. This inequality was improved by \textit{K.-D. Kirchberg} for Kähler manifolds [Ann. Global Anal. Geom. 4, 291-325 (1986; Zbl 0629.53058)]. Kähler manifolds of odd complex dimension satisfying the limiting case were classified by the author [\textit{A. Moroianu}, C. R. Acad. Sci., Paris, Sér. I 319, 1057-1062 (1994; Zbl 0851.58047)]. The present paper gives several results for the even-dimensional case, especially properties of the Ricci tensor of such a manifold.