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?

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 (CalabiYau), c) is dual to an ample line bundle  Fano case. In a) and b) there is always a KahlerEinstein 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 YauTianDonaldson conjecture. But all homogenious varieties are KahlerEinstein. 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 KahlerEinstein 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 HitchinThorpe 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 KahlerEinstein metric structure" this result can be found in Y. Matsushima. Remakrs on KahlerEinstein manifolds, Nagoya Math J. 46. (I found this reference in the book Besse, Einstein manifolds, 8.95). 


One of the most well known classes of K\"ahlerEinstein 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\"ahlerEinstein 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\"ahlerEinstein metric. We mention here two facts for isotropy irreducible Hermitian symmetric spaces $M=G/K$: 1) The isotropy subgroup $K$ has an 1dimensional 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. Moreever for any such complex structure we can define (a unique) homogeneous K\"ahlerEinstein 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\"ahlerEinstein metrics, Note that if some of the invariant complex structures are equivalent, then, the K\"ahlerEinstein metrics corresponding to these complex structures would be isometric. More information about the geometry of flag manifolds, painted Dynkin diagrams, K\"ahlerEinstein metrics, etc, can be found in the following article D. V. Alekseevsky: Flag manifolds, in Sbornik Radova, 11th Jugoslav. Geom. Seminar. Beograd 6 (14) (1997) 335. D. V. Alekseevsky and A. M. Perelomov: Invariant K\"ahlerEinstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986) 171182. 


If $X$ be a projective variety with positive kodaira dimension and the canonical ring $R(X,K_X)$ be finitely generated then even the first Chern class of $X$ do not have sign, then we have generalized Kahler Einstein metric on $X$, i.e. $$\pi:X\to X_{can}$$ we have $$Ric(\omega)=\omega+\omega_{WP}$$, where $\omega_{WP}$ is the WeilPetersson metric on moduli space of CalabiYau fibers. Since fibers $\pi$ are CalabiYau varieties by adjunction formula. 

