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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7$\begingroup$ 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. $\endgroup$– Torsten EkedahlCommented Aug 24, 2011 at 17:17
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5$\begingroup$ For the remaining questions, the short answer is no there are too many. See: mathoverflow.net/questions/59314/… $\endgroup$– Donu ArapuraCommented Aug 24, 2011 at 17:25
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1 Answer
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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.
answered Apr 24, 2016 at 1:27