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?
Which Kahler Manifolds Are Also Einstein Manifolds?
complex-geometrydg.differential-geometrykahler-manifolds
Best Answer
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).