Dear Common Crane,

let me list several results and question going in the direction of your intuition (all the information below I learned from Sergey Galkin)

1) Theorem. Hirzebruch-Kodaira, Yau. A Kahler manifold homeomorphic to $\mathbb CP^n$ is biholomorphic to $\mathbb CP^n$. There is a nice exposition of this result by Valentino Tossati: http://www.math.northwestern.edu/~tosatti/cpn.pdf

2) There is a question of Wilson:

If $V$ is a complex projective manifold of even dimension $n>k(V)$ and which has the same rational cohomology as $P^n$, is $V$ isomorphic to $P^n$?

3) In dimension 3 there are "four" complex projective manifolds with same Betti numbers as $\mathbb CP^3$ (these manifolds are $P^3$, the quadric, the manifold called $V_5$ and finally $V_{22}$. The first three manifolds are rigid, but the last one has six-dimensional moduli space of deformations).

4) Finally you can find some interesting recent development related to the question in the preprint http://member.ipmu.jp/sergey.galkin/papers/ipmu-11-0100.pdf where you will as well find references for articles in 1), 2).

3 deleted 12 characters in body

Dear Common Crane,

let me list several results and question going in the direction of your intuition (all the information below I learned from Sergey Galkin)

1) Theorem. Hirzebruch-Kodaira, Yau. A Kahler manifold homotopically equivalent homeomorphic to $\mathbb CP^n$ is biholomorphic to $\mathbb CP^n$.

2) There is a question of Wilson:

If $V$ is a complex projective manifold of even dimension $n>k(V)$ and which has the same rational cohomology as $P^n$, is $V$ isomorphic to $P^n$?

3) In dimension 3 there are "four" complex projective manifolds with same Betti numbers as $\mathbb CP^3$ (these manifolds are $P^3$, the quadric, the manifold called $V_5$ and finally $V_{22}$. The first three manifolds are rigid, but the last one has six-dimensional moduli space of deformations).

4) Finally you can find some interesting recent development related to the question in the preprint http://member.ipmu.jp/sergey.galkin/papers/ipmu-11-0100.pdf where you will as well find references for articles in 1), 2).

2 added 188 characters in body

Dear Common Crane,

let me list several results and question going in the direction of your intuition (all the information below I learned from Sergey Galkin)

1) Theorem. Hirzebruch-Kodaira, Yau. A Kahler manifold homotopically equivalent to $\mathbb CP^n$ is biholomorphic to $\mathbb CP^n$.

2) There is a question of Wilson:

*

If $V$ is a complex projective manifold of even dimension $n>k(V)$ and which has the same rational cohomology as $P^n$, is $V$ isomorphic to $P^n$?$P^n$?

3) In dimension 3 there are exactly four "four" complex projective manifolds with same Betti numbers as $\mathbb CP^3$ (these manifolds are $P^3$, the quadric, the manifold called $V_5$ and finally $V_{22}$. The first three manifolds are rigid, but the last one has six-dimensional moduli space of deformations).

4) Finally you can find some interesting recent development related to the question in the preprint http://member.ipmu.jp/sergey.galkin/papers/ipmu-11-0100.pdf where you will as well find references for articles in 1), 2).

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