Let $\mathbb CP^n$ denotes the complex projective space of dimension $n$, we have a standard complex structure of $\mathbb CP^n$, and my question is: is this complex structure unique?


Or equivalently, let $X$ be a complex manifold diffeomorphic to $\mathbb CP^n$, is $X$ biholomorphic to $\mathbb CP^n$?


What I know is from p45 of Morrow&Kodaira's book 《complex manifolds》:



> 
> $\mathbb CP^n$ is rigid.
> 
> 
> 


But this fact only ensures that small deformations don't change the complex structure of $\mathbb CP^n$, we did not even know whether the large deformations change the complex structure of $\mathbb CP^n$, or more generally, whether the same diffeomorphic type of $\mathbb CP^n$ admits different complex structures?


For dimension 1, I have learnt from some book that the answer is yes.  

For dimension 2, cited form Yau's 1977 paper 《Calabi's conjecture and some new results in algebraic geometry》, as a corollary of Yau's solution of Calabi's conjecture, the complex structure of $\mathbb CP^2$ is unique.


But for higher dimensions, is this problem solved? or any progress has been made?



Is the complex structure of $\mathbb CP^n$ unique?
Let me write this too long comment as an answer.


As abx says, what we do know is


**Theorem 1.** *If a Kähler manifold $X$ is homeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to it.*


This is due to Hirzebruch and Kodaira for $n$ odd (but with the strongest assumption for $X$ to be diffeomorphic to $\mathbb{CP}^n$, then relaxed to homeomorphic after work of Novikov), and to Yau for $n$ even.


For $n=2$, a stronger result holds, still proved by Yau, namely


**Theorem 2.** *If a compact complex surface $S$ is homotopy equivalent to $\mathbb{CP}^2$, then it is biholomorphic to it.*


In dimension $n\le 6$, we have instead a result due to Libgober-Wood which is stronger than Theorem 1, but weaker than Theorem 2, that states


**Theorem 3.** *A compact Kähler manifold of complex dimension $n\le 6$ which is homotopy equivalent to $\mathbb{CP}^n$ must be biholomorphic to it.*


You can find all this (and much more) on the very beautiful survey by V. Tosatti available [here](https://arxiv.org/pdf/1508.05641.pdf).


