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    $\begingroup$ It should be possible to show that majority of complex 3-folds are not Moishezon. So, I would not say that this remark is a real argument against existsing of a complex strucutre on S^6. There is a nice phrase in the aricle of Gromov. ihes.fr/~gromov/topics/SpacesandQuestions.pdf Page 30. "How much do we gain in global understanding of a compact (V, J) by assuming that the structure J is integrable (i.e. complex)? It seems nothing at all: there is no single result concerning all compact complex manifolds" $\endgroup$
    – Dmitri Panov
    Commented Jan 21, 2010 at 23:25
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    $\begingroup$ I'm not sure I understand this remark by Gromov. In the complex analytic case we have the Dolbeault resolution -- one of the ways to state the integrability condition is precisely that Dolbeault complex is a complex. This leads to topological statements, e.g. the alternating sum of the Euler characteristics of $\Omega^i$'s (computed using the Chern classes) is the Euler characteristic of the manifold itself. This may or may not be true in the almost-complex case, but I don't see how to prove it. $\endgroup$
    – algori
    Commented Jan 22, 2010 at 2:02
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    $\begingroup$ I think, the remark of Gromov is quite clear, it is quite hard to belive this remark, but the message is clear. As for Euler characteristics, David gave a correct explanation mathoverflow.net/questions/12601/… $\endgroup$
    – Dmitri Panov
    Commented Jan 22, 2010 at 9:05
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    $\begingroup$ What I meant was precisely that: this is hard to believe. The Euler characteristic is just the first thing that comes to mind. $\endgroup$
    – algori
    Commented Jan 22, 2010 at 17:41