Eigenvalue Estimates of the spinc Dirac Operator and Harmonic Forms on Kähler-Einstein Manifolds

Abstract

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 spinc structures. The limiting case is characterized by the existence of Kählerian Killing spinc 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 spinc spinor field vanishes. This extends to the spinc case the result of A. Moroianu stating that, on a compact Kähler-Einstein manifold of complex dimension 4ℓ+3 carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a Kählerian Killing spinor is zero.