The spectrum of the twisted Dirac operator on Kahler submanifolds of the   comlex projective space
  We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kahler submanifolds in Kahler manifolds carrying Kahlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding \CP^d \rightarrow \CP^n in order to test the sharpness of the upper bounds 
Nicolas Ginoux, Georges Habib
math.DG