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 :).

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. HirzebruchKodaira, 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 sixdimensional 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/ipmu110100.pdf where you will as well find references for articles in 1), 2). 

