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Is it a solved problem whether every smooth hypersurface X in CP^n admits a Kaehler-Einstein metric? Of course the c_1<0 and c_1 =0 cases are old, and every Fermat hypersurface does. It seems that existing knowledge implies they all do.

For example they are all Chow-Mumford stable. G. Tian then proved that the K-energy is proper, so if there are no hol. vector fields there is a K-E metric.

I'm reading someone's paper that has a result that gives another proof. But I'm not sure if this paper is correct yet.

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What is the paper? I would be surprised if it was already known that all hypersurfaces have a KE metric (at least this would be a very fresh result). On the other hand the following paper : arxiv.org/PS_cache/arxiv/pdf/0810/0810.1473v1.pdf together with Yau Tian Donladson conjecture would suggest that one should expect that the metric should exist. –  Dmitri Apr 20 '11 at 15:48
    
I don't think the paper that inspired me is posted on the arXiv yet. –  Craig Apr 21 '11 at 17:19

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