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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6$\begingroup$ 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). $\endgroup$– HenriCommented Mar 7, 2011 at 14:07
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$\begingroup$ Let $M$ is a Kähler manifold of dimension $2n$, then $M$ is Kähler-Einstein if and only if $\tau(L)=\tau(L^⊥)$ for all n-plane sections $L⊂T_pM$, where $\tau$ is the scalar curvature, Chen, Bang-Yen, Dillen, Franki Totally real bisectional curvature, Bochner-Kaehler and Einstein-Kaehler manifolds. Differential Geom. Appl. 10 (1999), no. 2, 145–154. $\endgroup$– user21574Commented Oct 23, 2017 at 21:48
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2 Answers
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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).
answered Mar 7, 2011 at 14:11
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$\begingroup$ 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 Riemann-Roch-Hirzebruch, 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 Kahler-Einstein! $\endgroup$– user21574Commented Oct 23, 2017 at 19:41
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$\begingroup$ Smyth showed that a complete complex hypersurface of $\mathbb CP^{n+1}$ is an Kahler-Einstein 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 n-dimensional complex quadric. see Smyth, B., 'Differential Geometry of Complex Hypersurfaces', Ann. Math. 85 (1967), 246- 266. Compare with result of S.T.Yau $\endgroup$– user21574Commented Oct 23, 2017 at 19:43
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One of the most well known classes of Kähler-Einstein 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 Kähler-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 structure is Kähler). It is well known that such a space $M=G/K$ admits a unique (as isotropy irreducible) Kähler-Einstein metric. Let me mention two basic 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.
A (generalized) flag manifold is a homogeneous Kähler manifold (the Kähler structure corresponds to the Kirillov-Kostant-Souriau 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 simply-connected 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 Kähler--Einstein metric, which is given in terms of the so-called Koszul form $$2\delta_{\frak{m}}=\sum_{\alpha\in R^{+}\backslash R_{K}^{+}}\alpha.$$ Thus, a flag manifolds admits a finite number of Kähler-Einstein metrics. Note that if some of the invariant complex structures are equivalent, then, the Kähler-Einstein metrics corresponding to these complex structures would be isometric.
More information about the geometry of flag manifolds, painted Dynkin diagrams, invariant Kähler-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 Kähler-Einstein metrics on compact homogeneous spaces, Funct. Anal. Appl. 20 (3) (1986) 171--182.
