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Is there a classification of the K\"ahler structure on the sphere? More generally, is there a classification of the K\"ahler 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
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For the remaining questions, the short answer is no there are too many. See: mathoverflow.net/questions/59314/… –  Donu Arapura Aug 24 '11 at 17:25
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