I am sure that there is an answer out there for the following question. If one is given an n dimensional Kahler manifold $X$ with Betti numbers that are the same as in the case of $\Bbb{CP}^n$, then is $X$ holomorphically diffeomorphic to $\Bbb{CP}^n$? This is of course true in the one dimensional case, but other then that I am clueless. If this is textbook stuff that I might have missed, references will be appreciated :).



Same betti numbers as $\Bbb{CP}^n$
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).


