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 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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