Eigenvalue Estimates of the ${\rm spin}^c$ Dirac Operator and Harmonic Forms on K\"ahler-Einstein Manifolds

TL;DR

This paper derives lower bounds for eigenvalues of the ${ m spin}^c$ Dirac operator on compact K"ahler-Einstein manifolds with positive scalar curvature, characterizing cases with special spinors and harmonic forms.

Contribution

It extends eigenvalue estimates and properties of K"ahlerian Killing ${ m spin}^c$ spinors to the ${ m spin}^c$ setting on K"ahler-Einstein manifolds.

Findings

Lower bounds for Dirac eigenvalues established.

Characterization of limiting cases via K"ahlerian Killing ${ m spin}^c$ spinors.

Clifford multiplication between harmonic forms and spinors vanishes.

Abstract

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 ${\rm spin}^c$ structures. The limiting case is characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing spinor is zero.