Is there any classification result(s) regarding how many symplectic structures on CP^n?
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One of the headline consequences of Taubes' work on SeibergWitten 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 holdsjust use the Moser trick. I'm not aware of any progress on this problem for $n>2$. 

