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replaced deprecated tag 'topology'; added tag 'open-problem' in view of the accepted answer
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Ricardo Andrade
Ricardo Andrade
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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?

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?

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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Hwang
Hwang
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Symplectic structures on a homotopy complex projective space

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?