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2D-Spheres with Complex Structure

  1. Sep 15, 2014 #1

    WWGD

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    Hi, it is a "well-known" result that the only 2d-sphere that allows

    a complex structure is S^2 ; it is open whether S^6 admits a complex

    structure, though it does admit an almost-complex structure. I know there

    are proofs that require knowledge of characteristics classes; does anyone know

    of proofs that do not require characteristic classes, or where knowledge of char. classes

    is not absolutely necessary for understanding the proof?

    Thanks.
     
  2. jcsd
  3. Sep 15, 2014 #2

    mathwonk

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    the proof of what?
     
  4. Sep 16, 2014 #3

    WWGD

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    Of the fact that S^4 , the 4-sphere does not admit a complex structure, and finding results about wether S^{2n} admits a complex structure.
     
  5. Sep 16, 2014 #4

    WWGD

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    One thing I know is that every Complex manifold admits a Symplectic structure, i.e , a closed non-degenerate 2-form w . Maybe someone knows of some homological obstruction to the existence of this form? I know we have that H^2(S^4)=0 , where H^2(S^4) is the 2nd homology group of the 4-sphere. I guess this means that every 2-form is exact. Does this make a difference?
     
  6. Sep 16, 2014 #5

    mathwonk

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