Roger Nakad, Mihaela Pilca, “Eigenvalue Estimates of the <nobr>${\mathop{\rm spin}^c}$</nobr> Dirac Operator and Harmonic Forms on Kähler–Einstein Manifolds”, SIGMA, 2015, Volume 11,054, 15 pp.

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

Roger Nakad^a, 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

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 ${\mathop{\rm spin}^c}$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\mathop{\rm spin}^c}$ 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 ${\mathop{\rm spin}^c}$ spinor field vanishes. This extends to the ${\mathop{\rm spin}^c}$ 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.

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

MSC: 53C27; 53C25; 53C55; 58J50; 83C60

Received: March 3, 2015; in final form July 2, 2015; Published online July 14, 2015

Language: English

DOI: 10.3842/SIGMA.2015.054