Eigenvalue Estimates of the ${\mathop{\rm spin}^c}$ Dirac Operator and Harmonic Forms on Kähler–Einstein Manifolds

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