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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 :).

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Even when $n=2$ there are 20-something isomorphism classes of smooth projective algebraic surfaces with the same Betti numbers as $\mathbb{CP}^2$. Look up "fake projective planes" and "fake projective spaces". –  David Hansen Jul 1 '12 at 6:05
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There are in fact exactly $100$ such surfaces (due to Prasad and Yeung); the first such was constructed by Mumford. They also have some results for $n>2$. –  ulrich Jul 1 '12 at 6:36
    
Oops, I read a mathscinet review too hastily. Thanks! –  David Hansen Jul 1 '12 at 7:02
    
I think the fake projective planes have torsion in homology, although this won't be detected by betti numbers. –  Ian Agol Nov 2 '12 at 22:04
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1 Answer 1

up vote 5 down vote accepted

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).

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