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For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.

My question is as follows. Let $M$ be a closed symplectic manifold which is homotopy equivalent to a complex projective space. Can we say that $M$ is homeomorphic or diffeomorphic to the standard one?

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up vote 9 down vote accepted

As far as I know your question is completely open. At the present moment no one knows if for $n>2$ there is a symplectic structure on any manifold homotopic to $\mathbb CP^n$ but not diffeomorphic to $\mathbb CP^n$. In fact our knowledge in these type of questions equals to zero. Namely, the following is open:

Question. Let $M^{2n}$ be any closed manifold ($n>2$) admitting an almost complex structure $J$ and a class $h\in H^2(M,\mathbb R)$ with $h^n\ne 0\in H^{2n}(M^{2n})$. Is it true that there is a symplectic form $w$ on $M^{2n}$ in the class $h$?

Your question about $\mathbb CP^{n}$ is open as well for $n=2$. We don't know yet if there exist manifolds homeomorphic to $\mathbb CP^2$ but not diffeomorphic to it. The only thing that can be said is that if such a symplectic $4$-fold exists it would be of general type.

But if I would bet, I would say that the answer to your question is no...

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