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Eigenvalue Estimates of the ${\mathop{\rm spin}^c}$ Dirac Operator and Harmonic Forms on Kähler–Einstein Manifolds
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SIGMA_2015_11_a53,
     author = {Roger Nakad and Mihaela Pilca},
     title = {Eigenvalue {Estimates} of the ${\mathop{\rm spin}^c}$ {Dirac} {Operator} and {Harmonic} {Forms} on {K\"ahler{\textendash}Einstein} {Manifolds}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2015},
     volume = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a53/}
}
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Roger Nakad; Mihaela Pilca. Eigenvalue Estimates of the ${\mathop{\rm spin}^c}$ Dirac Operator and Harmonic Forms on Kähler–Einstein Manifolds. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a53/

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