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?

Except for the $2$-sphere there are none, the cohomology class of the Kähler form is non-zero so for a Kähler manifold the second Betti number must be non-zero.
– Torsten EkedahlAug 24 '11 at 17:17