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When is a topological manifold which is not an almost complex manifold isomorphic to a real algebraic variety in the sense of locally ringed space?

It is well known that Serre's GAGA theorem solves complex analytic space and complex schemes over complex field.

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    $\begingroup$ A necessary condition is that manifold should be smooth. In this case, it is also sufficient by the Nash-Tognolli theorem. $\endgroup$ Dec 17, 2021 at 15:15
  • $\begingroup$ @DonuArapura Note that they are asking isomorphic "as locally ringed spaces", in which case the answer is, I suspect, never (e.g. because the Kähler differentials on a topological manifolds are always trivial). I'm not sure this is really what the OP wanted to ask though... $\endgroup$ Dec 17, 2021 at 22:42
  • $\begingroup$ @DenisNardin Right. I simply ignored that since I wasn't sure what was really intended $\endgroup$ Dec 17, 2021 at 23:27
  • $\begingroup$ complex manifold M, if it is complex analytical space, serre's GAGA thm solves . what i want to know for any topological manifold is isomorphic to a algebraic variety. $\endgroup$
    – user472602
    Dec 18, 2021 at 5:22

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