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    $\begingroup$ The existence of complex structure on $S^6$ is somewhat sensitive topic, but to my best knowledge the community consensus is that it's still very much an open problem. $\endgroup$
    – M.G.
    Commented Apr 14, 2021 at 15:20
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    $\begingroup$ You can take any almost complex structure which is not integrable and is algebraic, and then patch in local coordinates with a locally defined complex structure, using bump functions, so you get integrability on an open set, but not everywhere. It is easiest to see using complex linear coframings. $\endgroup$
    – Ben McKay
    Commented Apr 14, 2021 at 15:55
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    $\begingroup$ Take any symplectic manifold with $b_1$ odd, for example Thurston's example math.stonybrook.edu/~milivojevic/… This cannot be algebraic because $b_1$ would have to be even. Any symplectic manifold admits almost complex structure compatible in a rather nice way with the symplectic form. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 14, 2021 at 19:44
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    $\begingroup$ @LiviuNicolaescu: Every compact manifold is diffeomorphic to a real algebraic manifold. The question is not about complex algebraic manifolds, only real ones, so $b_1$ is not relevant. $\endgroup$
    – Ben McKay
    Commented Apr 15, 2021 at 9:49
  • $\begingroup$ @M.G. - can you specify "sensitive topic"? $\endgroup$
    – user122276
    Commented Apr 16, 2021 at 10:17