The classification of 4-dimensional Kähler manifolds with small eigenvalue of the Dirac operator
Let \(M^{2m}\) be a closed Kähler spin manifold and denote by \(R_ 0\) the minimum of the scalar curvature, \textit{K.-D. Kirchberg} [Ann. Global Anal. Geom. 4, 291-325 (1986; Zbl 0629.53058) and J. Geom. Phys. 7, No. 4, 449-468 (1990; Zbl 0734.53050)] proved the following estimations for the eigenvalues \(\lambda\) of the Dirac operator: \[ \lambda^ 2 \geq (m + 1)/4m\;R_ 0 \quad \text{ in case \(m\) is odd} \] \[ \lambda^ 2 \geq m/4(m-1)\;R_ 0\quad \text{ in case \(m\) is even}. \] The 6-dimensional and the 8-dimensional manifolds with \(\lambda^ 2 = 1/3\;R_ 0\) have been classified by \textit{K.-D. Kirchberg} [Math. Ann. 282, No. 1, 157-176 (1988; Zbl 0648.53040)] and \textit{A. Lichnerowicz} [C. R. Acad. Sci., Paris, Sér. I 311, No. 11, 717-722 (1990; Zbl 0713.53040)]. This classification is closely related to the classification of Kählerian twistor spaces and the manifolds under consideration are isometric to \(P^ 3(C)\) or \(F(1,2)\) \((m = 3)\) and \(P^ 3(C) \times T^ 2\) or \(F(1,2) \times T^ 2\) \((m = 4)\). In this paper we solve the corresponding classification problem in complex dimension \(m = 2\). It turns out that a 4-dimensional Kähler manifold \(M^ 4\) admits an eigenspinor to the eigenvalue \(\lambda^ 2 = 1/2\;R_ 0\) iff the scalar curvature is constant and \(H^ 0(M^ 4;K^{-1/2}) = 0\), where \(K\) denotes the canonical bundle. Using the classification of complex surfaces of Kodaira dimension \(-1\) we see that \(M^ 4\) is isomorphic to a ruled surface \(P(V)\) of a rank two bundle \(V\) over a Riemann surface \(F\). The cohomological condition as well as the Lichnerowicz-Calabi obstruction for Kähler metrics of constant scalar curvature yield that the genus of the Riemann surface \(F\) is bounded by one and the bundle \(V\) is trivial. Consequently, the space \(M^ 4\) is isometric to \(S^ 2 \times S^ 2\) or \(T^ 2 \times T^ 2\).
scientific article; zbMATH DE number 537225
scientific article