K. D. Kirchberg gave a lower bound for the first eigenvalue of the Dirac operator on a spin compact K"ahler manifold $M$ of odd complex dimension with positive scalar curvature. We prove that manifolds satisfying the limiting case are twistor space of quaternionic K"ahler manifold with positive scalar curvature.

We describe all simply connected ${Spin}^c$ manifolds carrying parallel and real Killing spinors. In particular we show that every Sasakian manifold (not necessarily Einstein) carries a canonical ${Spin}^c$ structure with Killing spinors.

We describe all compact spin K{"a}hler manifolds $(M^{2m},g,J)$ of even complex dimension and positive scalar curvature with least possible first eigenvalue of the Dirac operator. More precisely, we prove a result conjectured by Lichnerowicz, asserting that if there exists an eigenvalue $l$ of the Dirac operator on $M$ such that $lambda^2=frac{m}{4(m-1)}inf_M S$, (where $S$ the scalar curvature of $M$), then the universal cover of $M$ is isometric

It is proved that a compact K{"a}hler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two K{"a}hler-Einstein manifolds. A stronger result is established for the case of K"ahler surfaces. Without the compactness assumption, irreducible K"ahler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex