Abstract
In this paper Kählerian Killing spinors on manifolds of complex dimensionm=4l+3 are constructed. The construction is based on a theorem which states that a closed Kähler Einstein manifold of complex dimension 4l+3 and positive scalar curvature admits a Kählerian Killing spinor if and only if there is a complex (2l+1)-contact structure. In particular, any complex contact structure in the usual sense gives rise to such a generalized contact structure. Using this, new examples of Kählerian Killing spinors are obtained.
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References
[ABoSh]M.F. Atiyah, B. Bott, A. Shapiro, Clifford modules, Topology 3 (1964), 3–38.
[B]W.M. Boothby, Homogeneous complex contact manifolds, Proc. Symp. Pure Math. 3 (1961), 144–154.
[CFG]M. Cahen, A. Frank, S. Gutt, Spectrum of the Dirac operator on complex projective spaceP 2q−1(ℂ) Letters in Mathematical Physics 18 (1989), 165–176.
[FrK]Th. Friedrich, I. Kath, Compact seven-dimensional manifolds with Killing spinors, Comm. Math. Phys. 133 (1990), 543–561.
[Ki1]K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. Ann. Global Anal. Geom. 4 (1986), 291–325.
[Ki2]K.-D. Kirchberg, Compact six-dimensional Kähler spin manifolds of positive scalar curvature with smallest possible first eigenvalue of the Dirac operator. Math. Ann. 282 (1988), 157–176.
[Ki3]K.-D. Kirchberg, Killing spinors on Kähler manifolds, Ann. Global Anal. Geom. 11 (1993), 141–164.
[Ki4]K.-D. Kirchberg, Some further properties of Kälerian twistor spinors, Math. Nachr. 163 (1993), 229–255.
[KoO]S. Kobayashi, T. Ochiai, Characterization of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13:1 (1973), 31–47.
[Ko]S. Kobayashi, Remarks on complex contact manifolds, Proc. Amer. Math. Soc. 10 (1959), 164–167.
[M]A. Moroianu, La premiere valeur propre de l'operateur de Dirac sur les varietes kähleriennes compactes, to appear in Comptes-Rendus de l'Academie des Sciences de Paris.
[MSem]A. Moroianu, U. Semmelmann, Kählerian Killing Spinors, Complex Contact Structures and Twistor Spaces, Preprint 148 of E. Schrödinger Institute, Vienna, 1994.
[S]S.M. Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143–171.
[SeSem]S. Seifarth, U. Semmelmann, The spectrum of the Dirac operator on the complex projective spaceP 2q−1(ℂ), preprint, SFB 288, 1993.
[W]J.A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. and Mech. 14 (1965), 1033–1047.
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Kirchberg, K.D., Semmelmann, U. Complex contact structures and the first eigenvalue of the dirac operator on Kähler manifolds. Geometric and Functional Analysis 5, 604–618 (1995). https://doi.org/10.1007/BF01895834
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DOI: https://doi.org/10.1007/BF01895834