In the paper *On the complex projective spaces*, Hirzebruch and Kodaira prove the following: 



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> If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ if $n$ is odd or $n$ is even but $c\_1 \neq -(n+1)g$.
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Here, $c\_1$ is the first Chern class of $X$, and $g$ is a generator of $H^2(X,\mathbb{Z})$ with the same sign as the Kähler class.


My question is: Has the case $X$ is diffeomorphic but not biholomorphic to $\mathbb{CP}^n$ with $n$ even and $c\_1=-(n+1)g$ been ruled out in the following years? Or do we have some constructions of manifolds of this type?



On compact Kähler manifold diffeomorphic to complex projective space
The case when $n$ is even has been famously ruled out by Yau as a consequence of his proof of the Calabi Conjecture, see [his original paper](http://www.jstor.org/stable/67110) or [these notes](http://www.math.columbia.edu/%7Etosatti/cpn.pdf).


In fact, thanks to results of Novikov, you can replace diffeomorphism by homeomorphism and the statement remains true.


When $n=2$ in the same paper Yau proved that you can also drop the Kähler condition and replace homeomorphism by homotopy equivalence.


When $n>2$ the statement for Kähler manifolds but assuming only homotopy equivalence is true up to $n=6$ thanks to [Libgober and Wood](http://www.intlpress.com/JDG/archive/1990/32-1-139.pdf) (and true in general if you assume furthermore that the Pontryagin classes are the same as $\mathbb{CP}^n$), see also [this question](https://mathoverflow.net/questions/79630/fake-projective-spaces). The case of higher $n$ is probably open.


Finally, it is still an open problem to decide whether a compact complex manifold diffeomorphic to $\mathbb{CP}^n$ is biholomorphic to it (when $n\geq 3)$. If this were true, it would imply that the differentiable manifold $S^6$ does not admit a complex structure (otherwise, blowing up a point you'd get a complex manifold diffeomorphic but not biholomorphic to $\mathbb{CP}^3$), see for example [here](http://projecteuclid.org/euclid.dmj/1092749257).


