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Killing spinors on Kähler manifolds. (English) Zbl 0810.53033

Summary: In the paper Kählerian Killing spinors are defined and their basic properties are investigated. Each Kähler manifold that admits a Kählerian Killing spinor is Einstein of odd complex dimension. Kählerian Killing spinors are a special kind of Kählerian twistor spinors. Real Kählerian Killing spinors appear for example, on closed Kähler manifolds with the smallest possible first eigenvalue of the Dirac operator. For the complex projective spaces \(\mathbb{C} P^{2l - 1}\) and the complex hyperbolic spaces \(\mathbb{C} H^{2l - 1}\) with \(l > 1\) the dimension of the space of Kählerian Killing spinors is equal to \(\left ( \begin{smallmatrix} 2l\\ l\end{smallmatrix} \right)\). It is shown that in complex dimension 3 the complex hyperbolic space \(\mathbb{C} H^ 3\) is the only simple connected complete spin Kähler manifold admitting an imaginary Kählerian Killing spinor.

MSC:

53C20 Global Riemannian geometry, including pinching
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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