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Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?

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A theorem of Aubin-Yau says that any compact Kähler manifold with $c_1(X)$ negative or zero (as cohomology class) is Kähler-Einstein, in the sense that there exists $\omega$ a Kähler metric satisfyinf respectively $Ric(\omega)=-\omega$ (resp. $Ric(\omega)=0$). In the case where $c_1(X)>0$, it is not always true, but we know some examples (well-chosen weigted spaces e.g). –  Henri Mar 7 '11 at 14:07
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up vote 30 down vote accepted

This question can be interpreted in two different ways.

1) Which Kahler manifolds admit a Kahler metric that is at the same time Einstein?

2) Which Kahler manifolds admit an Einstein metric?

If you want 1), then you need to start with a manifold whose canonical bundle is either a) ample (like hypersurfaces of degree $\ge n+2$ in $\mathbb CP^n$), or b) trivial (Calabi-Yau), c) is dual to an ample line bundle - Fano case.

In a) and b) there is always a Kahler-Einstein metric by a theorem of Aubin and Yau. In the case c) we get a very subtle question, which is expected to be governed by Yau-Tian-Donaldson conjecture. But all homogenious varieties are Kahler-Einstein.

If you want 2), then the amount of Einstein metrics clearly becomes much larger. For example, $\mathbb CP^2$ blown up in one or two point do not admit a Kahler-Einstein metric, but they do admit an Einstein metric. For a reference to this statement you can check the article of Lebrun http://arxiv.org/abs/1009.1270 .

In general the question weather a given Kahler surface admits an Einstein metric is quite subtle. But at least there exists an obstruction. We can blow up any surface in sufficient number of points so that the obtained manifold violates Hitchin-Thorpe inequality http://en.wikipedia.org/wiki/Hitchin%E2%80%93Thorpe_inequality , hence not Einstein.

Finally, it was speculated (for example by Gromov here: http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf), that starting from real dimension 5 each manifold admits an Einstein metric.

Added reference. "Every compact, simply connected, homogeneous Kahler manifold admits a unique (up to homothety) invariant Kahler-Einstein metric structure"- this result can be found in Y. Matsushima. Remakrs on Kahler-Einstein manifolds, Nagoya Math J. 46. (I found this reference in the book Besse, Einstein manifolds, 8.95).

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One of the most well known classes of K\"ahler-Einstein manifolds, i.e. manifolds which carry a Kahler metric $g$ such that $Ric_{g}= \lambda \cdot g$ $c\in\mathbb{R}$, are the generalized flag manifolds $$G^{\mathbb{C}}/P\cong G/K$$ of a compact connected simple Lie group. Here $P$ is a parabolic subgroup of the complexification $G^{\mathbb{C}}$ of $G$, and $K=p\cap G$ is the centralizer of a torus $S\subset G$, i.e. $K=C(S)$. If $S=T=$maximal torus, then we obtain a full flag manifold $G/T$.

In the family of generalized flag manifolds, we find a very important subfamily of K\"ahler-Einstein manifolds, the (isotropy irreducible) Hermitian symmetric spaces $M=G/K$ of compact type (i.e. compact symmetric spaces endowed with a Hermitian structure invariant under the symmetries. In particular, this Hermitian structire is Kahler) . It well know that such a space $M=G/K$ admits a unique (as isotropy irreducible) K\"ahler-Einstein metric. We mention here two facts for isotropy irreducible Hermitian symmetric spaces $M=G/K$:

1) The isotropy subgroup $K$ has an 1-dimensional center.

2) They are the only generalized flag manifolds which are the same time symmetric spaces.

Now, a (generalized) flag manifold is also a homogenepous K\"ahler manifold (the K\"ahler structure on these manifolds arising form the fact that any flag manifold is an adjoint orbit of an element in the Lie algebra of $G$). In particular, flag manifolds exhaust all compact simply connected Homogeneous Kahler manifolds corresponding to (compact, connected) simple Lie groups. Their classification is based on the painted Dynkin diagrams.

Now, any $M=G^{\mathbb{C}}/P=G/C(S)$ admits a finite number of invariant complex structures. More-ever for any such complex structure we can define (a unique) homogeneous K\"ahler--Einstein metric, which given in terms of representation theory by computing the Koszul form $2\delta_{\frak{m}}$. Thus a flag manifolds admits a finite number of K\"ahler-Einstein metrics, Note that if some of the invariant complex structures are equivalent, then, the K\"ahler-Einstein metrics corresponding to these complex structures would be isometric.

More information about the geometry of flag manifolds, painted Dynkin diagrams, K\"ahler-Einstein metrics, etc, can be found in the following articles:

D. V. Alekseevsky: Flag manifolds, in Sbornik Radova, 11th Jugoslav. Geom. Seminar. Beograd 6 (14) (1997) 3--35.

D. V. Alekseevsky and A. M. Perelomov: Invariant K\"ahler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986) 171--182.

Y. Sakane: Homogeneous Einstein metrics on flag manifolds, Lobachevskii J. Math. (4) (1999) 71--87.

M. Kimura: Homogeneous Einstein metrics on certain K\"ahler C-spaces, Adv. Stud. Pure Math. 18-I (1990) 303--320.

M. Bordeman, M. Forger and H. R\"omer: Homogeneous K\"ahler manifolds: paving the way towards new supersymmetric sigma models, Comm. Math. Phys. 102 (1986) 604--647.

A. Borel and F. Hirzebruch: Characteristics classes and homogeneous spaces I, Amer. J. Math. 80 (1958) 458--538.

A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on generalized flag manifolds with two isotropy summands, J. Aust. Math. Soc. 90 (2) (2011) 237--251.

A. Arvanitoyeorgos and I. Chrysikos: Invariant Einstein metrics on generalized flag manifolds with four isotropy summands, Ann. Glob. Anal. Geom. 37 (2) (2010) 185-219.

M. Nishiyama: Classification of invariant complex structures on irreducible compact simply connected coset spaces, Osaka J. Math. 21 (1984) 39--58.

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