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