Remarks on Kähler-Einstein Manifolds

Abstract

The main purpose of this note is to characterize a compact KahlerEinstein manifold in terms of curvature form. The curvature form Ω is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Kahler metric is the harmonic representative of the curvature class if and only if the Kahler metric is an Einstein metric in the generalized sense {g.s.) y that is, if the Ricci form of the metric is parallel. It is well known that a Kahler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kahler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Kahler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kahler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger [3] in the case of a complex projective space and our proof is completely different from Berger's.1. Throughout this paper we shall denote by M a compact Kahler manifold and by T and Γ* the holomorphic tangent bundle and the holomorphic cotangent bundle of M respectively. The real differentiable tangent bundle of M will be denoted by T& The vector space of smooth sections of a vector bundle F will be denoted by Γ(F). A section X of T is a complex vector field of holomorphic type or of type (1,0) and we denote by X the conjugate of X; X is a section of the conjugate bundle T of T. We denote by <, > the Hermitian metric in 7\ that is, if X, Y e Γ(T) f then <X, Y> = g{X, Ϋ),where g denotes the Kahler metric in M. We have then

On the spectrum of the Page and the Chen–LeBrun–Weber metrics

“…A heuristic reason for why on might expect s −1 k to give a better bound than s k comes from examining what happens in the Kähler-Einstein case. Here one has the classical estimate due to Matsushima [17] and later generalised by Lichnerowicz [16]. We use the version stated in [3].…”

Les vari�t�s de dimension 4 � signature non nulle dont la courbure est harmonique sont d'Einstein

“…L'attention avait d6j/t 6t6 attir6e dans le pass6 sur la famille des mOtriques gt courbure harmonique (par exemple, voir [17] page 1 et [18] page 10 et [20]). …”

“…Corollaire 3.t4 et Proposition 3.17). Matsushima a justement montr6 dans [20] que toute mdtrique k~ihl6rienne /t courbure harmonique 6tait localement un produit de m6triques d'Einstein-K~ihler (pour une autre preuve, voir [10]). L'hypoth6se sur la signature est n6cessaire puisque Derdzinski a donn6 dans [9] des exemples de mhtriques 5. courbure harmonique et 5. courbure de Ricci non parall4le sur la variht6 S~x S 3 (en fait ces m4triques sont conformhment plates 5. courbure scalaire constante).…”

Compact K�hler manifolds with nonnegative sectional curvature

“…Corollary 1.3 is a theorem of M. Berger [3] as generalized by Bishop and Goldberg [5][6][7]. See also Matsushima [18]. In fact, our Theorem 1.1 can be viewed as a generalization of Corollary 1.3.…”