2D-Spheres with Complex Structure

[WWGD]

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.

 

[mathwonk]

the proof of what?

 

[WWGD]

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.

 

[WWGD]

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?

 

[mathwonk]

http://mathoverflow.net/questions/11664/complex-structure-on-sn