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?