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

Research paper by André Lichnerowicz

Indexed on: 01 Oct '98Published on: 01 Oct '98Published in: Letters in Mathematical Physics


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 dimC W = 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.