In a joint work with Mikaela Pilca (University of Regensburg-Germany), we establish a lower bound for the first eigenvalue of the Spin Dirac operator defined on a Kähler-Einstein manifold of positive scalar curvature. This lower bound involves the index of , its scalar curvature and an integer defining the Spin structure. The limiting case is characterized by the existence of special spinor fields called Kählerian Killing spinors. As a geometric application of the limiting case, we prove that the only harmonic complex forms of type () on Kähler-Einstein manifolds admitting a complex contact structure are the constant multiples of the Kähler form.
