MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 :).

share|cite|improve this question
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
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
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:

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 where you will as well find references for articles in 1), 2).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.