Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz
Using the well-known Lichnerowicz formula for the square of the Dirac operator, Th. Friedrich proved a first inequality for the eigenvalues of the Dirac operator on a compact Riemannian spin manifold. He obtained a lower bound which contains the scalar curvature of the underlying manifold. The limiting case of this inequality is characterized by the existence of real Killing spinors. In 1986 K.-D. Kirchberg improves this inequality for Kähler manifolds. Kähler manifolds of odd complex dimension satisfying the limiting case were classified by the author in 1994. In 1997, the author gave several partial results for the even-dimensional case, especially properties of the Ricci tensor of such a manifold. The paper under review gives a complete description for all compact spin Kähler manifolds of even complex dimension and positive scalar curvature with least possible first eigenvalue of the Dirac operator.
scientific article; zbMATH DE number 1355994
scientific article