A positively curved compact Kahler manifold has second Betti number equal to 1 by a result of Bishop and Goldberg.  So as Torsten Ekedahl commented, the only sphere that admits a Kahler structure is a 2-sphere.

And as Donu Arapura commented, a discussion of the classification of Kahler structures is [here][1].  For another discussion, see [here][2].


  [1]: https://mathoverflow.net/questions/59314/kahler-metrics-for-projective-space-that-are-not-the-fubini-study-metric
  [2]: https://mathoverflow.net/questions/178494/all-k%C3%A4hler-metrics-on-a-complex-manifold?rq=1