First Eigenvalue of the Dirac Operator for a Kähler Manifold of Even Complex Dimension: The Limiting Case

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K. D. Kirchberg has given a minoration of the absolute value of the eigenvalues of the Dirac operator for a compact Kähler spin manifold (W, g) with positive scalar curvature and has introduced, in this context, the notion of Kähler twistor-spinor. We prove here that if dimℂW = p ≥ 4 is even, in the limiting case, (W, g) is the Kähler product of an odd-dimensional limiting case compact Kähler spin manifold of complex dimension (p–1), by a flat Kähler manifold which is a compact quotient of ℂ.

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