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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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    $\begingroup$ 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. $\endgroup$
    – Jason Starr
    Commented Sep 1, 2013 at 12:40
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    $\begingroup$ A recent article by X. Chen, S. Sun and S. Donaldson www2.imperial.ac.uk/~skdona/KEDEVELOPMENTS-9-19-2013.PDF $\endgroup$
    – user39400
    Commented Nov 21, 2013 at 2:51
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    $\begingroup$ 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? $\endgroup$
    – user43004
    Commented Nov 21, 2013 at 4:25
  • $\begingroup$ @user43004: As I understand it, stability is purely algebraic, so we can check it explicitly in many algebraic varieties, if we can write down algebraic descriptions of the varieties. KE metrics, although somehow more comfortable for differential geometers, and admitting deep consequences, are not explicitly written down on typical KE manifolds. So proving stability is often easier than proving KE. $\endgroup$
    – Ben McKay
    Commented Nov 19, 2018 at 19:09

1 Answer 1

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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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    $\begingroup$ 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. $\endgroup$
    – Ruadhaí Dervan
    Commented Aug 7, 2013 at 17:55
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    $\begingroup$ Its also worth noting that as far as I understand, Tian has also posted a preprint with these results: arXiv:1211.4669. $\endgroup$
    – Otis Chodosh
    Commented Aug 7, 2013 at 23:30
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    $\begingroup$ 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, $\endgroup$
    – diverietti
    Commented Sep 2, 2013 at 11:54
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    $\begingroup$ Might be worth recording here that Chen-Donaldson-Sun were awarded the Veblen prize for their work: ams.org/news?news_id=4705. $\endgroup$
    – Sam Hopkins
    Commented Nov 19, 2018 at 18:13

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