The classification of 4-dimensional Kähler manifolds with small eigenvalue of the Dirac operator
type: Item
instance of: scholarly article
title: The classification of 4-dimensional Kähler manifolds with small eigenvalue of the Dirac operator
zbMATH Open document ID: 0798.53065
DOI: 10.1007/BF01444903
author: Thomas Friedrich
published in: Mathematische Annalen
publication date: 1994-11-01T00:00:00Z
full work available at URL: https://eudml.org/doc/165058
review text: 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 \(T^ 2 \times T^ 2\).
reviewed by: Thomas Friedrich
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 537225
zbMATH Keywords: Kähler spin manifold
zbMATH Keywords: scalar curvature
zbMATH Keywords: eigenvalues
zbMATH Keywords: Dirac operator
zbMATH Keywords: twistor spaces
zbMATH Keywords: eigenspinor
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: ruled surface
MaRDI profile type: Publication
cites work: Der erste Eigenwert des Dirac‐Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung
cites work: Compact four-dimensional self-dual EINSTEIN manifolds with positive scalar curvature
cites work: Remarks on extremal Kähler metrics on ruled manifolds
cites work: Q4195061
cites work: Kählerian Twistor Spaces
cites work: An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature
cites work: Compact six-dimensional Kähler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator
cites work: The first eigenvalue of the Dirac operator on Kähler manifolds
cites work: Properties of Kählerian twistor-spinors and vanishing theorems
cites work: Q3198480
cites work: Q3200188
cites work: Spektren symmetrischer Räume
cites work: On ruled surfaces of genus 1
cites work: On the curvature of compact Hermitian manifolds
description: scientific article; zbMATH DE number 537225