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    $\begingroup$ This is interesting. Doesn't this bear some similarity to the argument used by Adler (who is or was a colleague of LeBrun) in his published "proof" of this conjecture? My recollection is that Adler tried to show that a Riemannian metric compatible with a complex structure on S^6 could be deformed into a Kahler metric, leading to the same contradiction. By the way, I never found anyone who was able to identify exactly why Adler's proof is wrong. $\endgroup$
    – Deane Yang
    Commented Oct 26, 2009 at 1:13
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    $\begingroup$ That's right. He has a continuity argument involving a notion of "distinguished metric" for an almost complex structure that I have some difficulty making sense of: it requires embedding yr almost complex manifold in some high dimensional sphere. $\endgroup$
    – Fran Burstall
    Commented Oct 26, 2009 at 21:10
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    $\begingroup$ Yes, I got completely lost when he embedded the sphere into a high dimensional space. I couldn't see why that would help at all and the calculations become a total mess. $\endgroup$
    – Deane Yang
    Commented Aug 8, 2013 at 17:29