In the paper 'On the complex projective spaces' of Hirzbruch and Kodaira, they prove that

If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ in case $n$ 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 of the Kähler class.

My question is: Has the case $X$ is diffeomorphic to $\mathbb{CP}^n$, $n$ even and $c_1=-(n+1)g$ been ruled out in the following years? Or we have some constructions of manifolds of this type?

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?

In the paper 'On the complex projective spaces' of Hirzbruch and Kodaira,they prove that

If $X$ is compact Kahler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ in case $n$ 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 $H^2(X,Z)$ with the same sign of the Kahler class.

My question is  :Has the case X is diffeomorphic to $\mathbb{CP}^n$  ,$n$ even and $c_1=-(n+1)g$ been ruled out in the following years?Or we have some constructions of manifolds of this type?

In the paper 'On the complex projective spaces' of Hirzbruch and Kodaira, they prove that

If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ in case $n$ 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 of the Kähler class.

My question is: Has the case $X$ is diffeomorphic to $\mathbb{CP}^n$,  $n$ even and $c_1=-(n+1)g$ been ruled out in the following years? Or we have some constructions of manifolds of this type?

In the paper 'On the complex projective spaces' of Hirzbruch and Kodaira,they prove that

 If $X$ is compact Kahler manifold diffeomorphic to $\mathbb{CP}^n$,then $X$ is biholomorphic to $\mathbb{CP}^n$ in case $n$ 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 $H^2(X,Z)$ with the same sign of the Kahler class.

My question is :Has the case X is diffeomorphic to $\mathbb{CP}^n$ ,$n$ even and $c_1=-(n+1)g$ been ruled out in the following years?Or we have some constructions of manifolds of this type?

In the paper 'On the complex projective spaces' of Hirzbruch and Kodaira,they prove that

If $X$ is compact Kahler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{CP}^n$ in case $n$ 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 $H^2(X,Z)$ with the same sign of the Kahler class.

My question is :Has the case X is diffeomorphic to $\mathbb{CP}^n$ ,$n$ even and $c_1=-(n+1)g$ been ruled out in the following years?Or we have some constructions of manifolds of this type?