Is every complex manifold that's homeomorphic to $\mathbb C\mathbb P^n$ also isomorphic to $\mathbb C\mathbb P^n$?

Question

Put another way, is the complex structure on $\mathbb C\mathbb P^n$ unique? I know that this is the case for $n\in\{1,2\}$, so I'm curious as to whether it's the case in general. If it's not known, is it believed to be true, and what progress has been made on this question?

Answer 1

The following theorem is due to Hirzebruch, Kodaira, and Yau:

Theorem: Let $M$ be a compact Kähler manifold homeomorphic to $\mathbb{CP}^n$. Then $M$ is biholomorphic to $\mathbb{CP}^n$.

Moreover, Libgober and Wood showed that the result remains true for $n \leq 6$ if $M$ is only assumed to be homotopy equivalent to $\mathbb{CP}^n$.

This still leaves open the possibility that there is a compact complex manifold $M$ homeomorphic to $\mathbb{CP}^n$ which is not biholomorphic to it, but such an $M$ is necessarily non-Kähler. For $n = 1$ and $2$, this is known not to occur, but is open for $n \geq 3$. In particular, if $S^6$ admits a complex structure, then there is a compact complex manifold diffeomorphic to $\mathbb{CP}^3$ which is not biholomorphic to it.

For details on all of these results, see the excellent survey Uniqueness of $\mathbb{CP}^n$ by Valentino Tosatti.