Eigenvalue Estimates of the ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ Dirac Operator and Harmonic Forms on K\"ahler-Einstein Manifolds

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ structures. The limiting case is characterized by the existence of K"ahlerian Killing ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K"ahlerian Killing ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ spinor field vanishes. This extends to the ${\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ case the result of A. Moroianu stating that, on a compact K"ahler-Einstein manifold of complex dimension $4\mathrm{\ell }+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K"ahlerian Killing spinor is zero.