# On compact Kähler manifold diffeomorphic to complex projective space

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

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 $c_1 \neq -(n+1)g$.

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?

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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 or these notes.
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 (and true in general if you assume furthermore that the Pontryagin classes are the same as $\mathbb{CP}^n$), see also this question. 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.