Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 054, 15 pages      arXiv:1502.05252      https://doi.org/10.3842/SIGMA.2015.054

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

Roger Nakad a and Mihaela Pilca bc
a) Notre Dame University-Louaizé, Faculty of Natural and Applied Sciences, Department of Mathematics and Statistics, P.O. Box 72, Zouk Mikael, Lebanon
b) Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, 93040 Regensburg, Germany
c) Institute of Mathematics ''Simion Stoilow'' of the Romanian Academy, 21, Calea Grivitei Str, 010702-Bucharest, Romania

Received March 03, 2015, in final form July 02, 2015; Published online July 14, 2015

Abstract
We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian 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 Kählerian 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 Kähler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a Kählerian Killing spinor is zero.

Key words: ${\rm spin}^c$ Dirac operator; eigenvalue estimate; Kählerian Killing spinor; parallel form; harmonic form.

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