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I heard in a conference that Yau's conjecture is open for positive Chern class. I read in an article that talked about some stability conditions necessary in this case. So I want to know if this stability condition is well-determined or still conjectural. More precisely, is there any precise statement of this conjecture or is it open ended?

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I would like to call the attention of the moderators of MO. What started as a good question and informative answer now seems to have been hijacked by (apparently) anonymous users to argue one side of a dispute. Without taking any position on that dispute, I do not feel MO is the proper place for such arguments. –  Jason Starr Sep 1 '13 at 12:40
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A recent article by X. Chen, S. Sun and S. Donaldson www2.imperial.ac.uk/~skdona/KEDEVELOPMENTS-9-19-2013.PDF –  user39400 Nov 21 '13 at 2:51
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can anyone tell me the point of proving KE by stability? It seems that stability is a harder thing compared with KE but why use a complicated thing to explain an easier concept of KE? –  geometrican Nov 21 '13 at 4:25
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1 Answer

up vote 7 down vote accepted

A precise statement and proof of the relationship between stability and the existence of Calabi-Yau metrics is in:

  1. arXiv:1302.0282, Xiuxiong Chen, Simon Donaldson, Song Sun, Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof

  2. arXiv:1212.4714, Xiuxiong Chen, Simon Donaldson, Song Sun, Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π

  3. arXiv:1211.4566, Xiu-Xiong Chen, Simon Donaldson, Song Sun, Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities

  4. arXiv:1210.7494, Xiu-Xiong Chen, Simon Donaldson, Song Sun Kahler-Einstein metrics and stability

  5. arXiv:1211.4669, Gang Tian, K-stability and Kähler-Einstein metrics

(so I understand; I haven't read all of this)

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It's worth noting that the above preprints prove that K-stable implies the existence of a Kahler-Einstein metric. The other direction was proven when there are no automorphisms by Donaldson (KE implies K-semistability), Stoppa (KE implies K-stability) and in the presence of automorphsims by Berman. –  Ruadhaí Dervan Aug 7 '13 at 17:55
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Its also worth noting that as far as I understand, Tian has also posted a preprint with these results: arXiv:1211.4669. –  Otis Chodosh Aug 7 '13 at 23:30
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Yes, maybe for the sake of completeness, and since moreover your answer has been accepted, you could add also Tian's contribution to the subject. Best, –  diverietti Sep 2 '13 at 11:54
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