# Surface with Kahler-Einstein metric

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 $$k=3, 4$$.

• Perhaps you should add simply connected. Otherwise Enriques surfaces satisfy all the above, with k=9 Jan 20, 2019 at 11:01
• @MartindeBorbon Thanks for the comment. I think I understand what you said. And it justifies why I put the condition $3\leq k\leq 8$ in the beginning.
– Tong
Jan 20, 2019 at 11:20
• Right, but I still think you should add the hypothesis that M is simply connected. Otherwise blow ups of fake projective spaces at k points are counter examples Jan 20, 2019 at 11:52
• I am trying to understand what you said....are you saying the blow ups of fake projective spaces have Kahler-Einstein metric?
– Tong
Jan 20, 2019 at 12:01
• my mistake, no these are not, the canonical is not ample Jan 20, 2019 at 12:48

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
• I see. I wonder if the statement is true for $k=3,4$. I will try to wait and see if there is any result for $k=3,4$. If not, I will accept your answer. Thanks.