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?

Kählermanifold homeomorphic to $\mathbb{P}^n$ is isomorphic to $\mathbb{P}^n$ — this follows from Yau's theorem, plus some preious work of Kobayashi-Ochiai. $\endgroup$1more comment