Is there any classification result(s) regarding how many symplectic structures on CP^n?
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$\begingroup$ Related question: mathoverflow.net/questions/59314/… $\endgroup$– José Figueroa-O'FarrillCommented Aug 31, 2011 at 0:41
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One of the headline consequences of Taubes' work on Seiberg-Witten theory on symplectic four- manifolds was that the standard symplectic form on $\mathbb{C}P^2$ is the unique one (up to scale, of course); see Theorem B of this paper. Of course for $\mathbb{C}P^1$ the same result holds---just use the Moser trick.
I'm not aware of any progress on this problem for $n>2$.
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1$\begingroup$ Up to scale and up to symplectomorphism, is what you meant to say, of course. Because if $\omega$ is closed and non-degerate, then adding a sufficiently small exact $2$-form will preserve these two conditions. $\endgroup$ Commented Aug 31, 2011 at 11:13