Let $3\leq k\leq 8$ be an integer. Suppose $M$ is a complex surface which has a Kahler-Einstein metric and has the same Betti numbers as $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$, i.e. 
$$b\_0(M)=b\_4(M)=1,~~ b\_1(M)=b\_3(M)=0,~~b\_2^+(M)=1,~~ b\_2^-(M)=k.$$
Then can we conclude that $M$ is biholomorphic to $\mathbb{C}\mathbb{P}^2\# k\overline{\mathbb{C}\mathbb{P}^2}$?


**Note added**: The answer of Martin de Borbon has shown that this is not true for $5\leq k\leq 8$. I would like to know if the statement is true for $k=3, 4$. 



Surface with Kahler-Einstein metric
There answer is no. The topological manifolds $\mathbb{CP}^2 \sharp k \overline{\mathbb{CP}^2}$ admit smooth structures that support KE with negative scalar curvature for k=5, 6, 7, 8


For k=8, see <https://arxiv.org/pdf/dg-ga/9705007.pdf> 


For k=6, 7, see <https://arxiv.org/pdf/0806.1424.pdf>


For k=5, see <https://msp.org/gt/2009/13-3/gt-v13-n3-p06-p.pdf>


