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

TL;DR

This paper derives upper bounds for small eigenvalues of the twisted Dirac operator on Kahler submanifolds and computes the spectrum for the canonical embedding of complex projective spaces to evaluate the bounds.

Contribution

It provides new upper estimates for eigenvalues of the twisted Dirac operator on Kahler submanifolds and explicitly computes the spectrum for a key embedding.

Findings

Upper bounds for small eigenvalues established

Spectrum computed for the canonical embedding of \\CP^d into \\CP^n

Results test the sharpness of the derived bounds

Abstract

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