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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?

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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 Ekedahl Aug 24 '11 at 17:17
For the remaining questions, the short answer is no there are too many. See:… – Donu Arapura Aug 24 '11 at 17:25

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. For another discussion, see here.

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