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?

6$\begingroup$ A theorem of AubinYau says that any compact Kähler manifold with $c_1(X)$ negative or zero (as cohomology class) is KählerEinstein, 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 (wellchosen weigted spaces e.g). $\endgroup$– HenriMar 7, 2011 at 14:07

$\begingroup$ Let $M$ is a Kähler manifold of dimension $2n$, then $M$ is KählerEinstein if and only if $\tau(L)=\tau(L^⊥)$ for all nplane sections $L⊂T_pM$, where $\tau$ is the scalar curvature, Chen, BangYen, Dillen, Franki Totally real bisectional curvature, BochnerKaehler and EinsteinKaehler manifolds. Differential Geom. Appl. 10 (1999), no. 2, 145–154. $\endgroup$– user21574Oct 23, 2017 at 21:48
2 Answers
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).

$\begingroup$ Let $M$ is an algebraic hypersurface of degree $d$ imbedded in $CP^{n+1}$ with the Kaehler metric induced from $CP^{n+1}$, then, by a theorem of RiemannRochHirzebruch, the first Chern class of $M$ is given by $c_1(M) = ((n + 2  d)/4\pi)[\omega]$, BUT this never say that they are KahlerEinstein! $\endgroup$– user21574Oct 23, 2017 at 19:41

$\begingroup$ Smyth showed that a complete complex hypersurface of $\mathbb CP^{n+1}$ is an KahlerEinstein with respect to the Kaehler metric induced from $CP^{n+1}$ if and only if it is either a complex hyperplane $\mathbb CP^n$ or an ndimensional complex quadric. see Smyth, B., 'Differential Geometry of Complex Hypersurfaces', Ann. Math. 85 (1967), 246 266. Compare with result of S.T.Yau $\endgroup$– user21574Oct 23, 2017 at 19:43
One of the most well known classes of KählerEinstein manifolds, i.e. complex manifolds which carry a Kähler 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 $G/T$.
Inside the class of generalized flag manifolds, we find a very important subclass of KählerEinstein 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 structure is Kähler). It is well known that such a space $M=G/K$ admits a unique (as isotropy irreducible) KählerEinstein metric. Let me mention two basic 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.
A (generalized) flag manifold is a homogeneous Kähler manifold (the Kähler structure corresponds to the KirillovKostantSouriau symplectic form, since any flag manifold can be viewed as an adjoint orbit of an element in the Lie algebra of $G$). In particular, flag manifolds exhaust all compact and simplyconnected homogeneous Kähler manifolds $M=G/K$ corresponding to a compact, connected, simple Lie group $G$. Their classification is based on the painted Dynkin diagrams.
Any coset $M=G^{\mathbb{C}}/P=G/K$ $(K=C(S))$ admits a finite number of invariant complex structures. Moreover, for any such complex structure we can define (a unique) homogeneous KählerEinstein metric, which is given in terms of the socalled Koszul form $$2\delta_{\frak{m}}=\sum_{\alpha\in R^{+}\backslash R_{K}^{+}}\alpha.$$ Thus, a flag manifolds admits a finite number of KählerEinstein metrics. Note that if some of the invariant complex structures are equivalent, then, the KählerEinstein metrics corresponding to these complex structures would be isometric.
More information about the geometry of flag manifolds, painted Dynkin diagrams, invariant KählerEinstein 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) 335.
D. V. Alekseevsky and A. M. Perelomov: Invariant KählerEinstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986) 171182.