Is there a classification of the Kähler structure on the sphere? More generally, is there a classification of the Kähler structures on the complex projective spaces? Even more generally, what about the flag manifolds?
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.