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We establish an upper estimate for the small eigenvalues of the twisted Diracoperator on Kahler submanifolds in Kahler manifolds carrying Kahlerian Killingspinors. We then compute the spectrum of the twisted Dirac operator of thecanonical embedding \CP^d \rightarrow \CP^n in order to test the sharpness ofthe upper bounds

The spectrum of the twisted Dirac operator on Kahler submanifolds of the comlex projective space