The spectrum of the twisted Dirac operator on Kähler submanifolds of the complex projective space

by Nicolas Ginoux and Georges Habib

manuscripta math. 137 (2012), no. 1-2, 215-231

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Abstract: We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kähler submanifolds in Kähler manifolds carrying Kählerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding of the d-dimensional complex projective space into the n-dimensional one in order to test the sharpness of the upper bounds.

After publishing this article, we were informed by Seán Murray that the computation of the spectrum of the Dirac operator for any spinc structure on the complex projective space of arbitrary complex dimension had been first done in:

B. P. Dolan, I. Huet, S. Murray, D. O'Connor, A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces, J. High Energy Phys. no. 3 (2008), 029, 21 pp.

In the preliminary version of their article (, the eigenvalues are given by the identity (50) p. 10 and the multiplicities of the non-vanishing eigenvalues by the identity (51) p. 10 (resp. identity (22) p.5 for the possible eigenvalue 0). To compare their identity (50) with ours (Theorem 4.7 p. 228), replace their $q$ by $\frac{d+1}{2}-m$ and notice that the spectrum does not change when replacing $q$ by $d+1-q$.

Nicolas Ginoux, 3.03.2012