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An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. (English) Zbl 0629.53058
The main result of the paper is the following Theorem: Let $$\lambda$$ be an eigenvalue of the Dirac operator D on a closed Kähler spin manifold $$M^ n$$ of real dimension n with positive scalar curvature R, then $(*)\quad \lambda^ 2\geq ((n+2)/n)R_ 0/4$ where $$R_ 0$$ denotes the minimum of R on $$M^ n$$. If $$\lambda_ 0:=\sqrt{(n+2)R_ 0/4n}$$ itself is an eigenvalue of D, then $$M^ n$$ is an Einstein space of odd complex dimension $$m=n/2.$$
The estimate (*) is sharp in the sense that there exist Kähler manifolds for which $$\lambda_ 0$$ itself is an eigenvalue of D. In dimension $$n=6$$ the complex projective space $$P^ 3({\mathbb{C}})$$ and the flag manifold $$F({\mathbb{C}}^ 3)$$ with respect to their natural Kähler structures are examples of such spaces. Later we have shown in another paper that for $$n=6$$ there are no other examples.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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##### References:
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