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 $k=3, 4$.

  • $\begingroup$ Perhaps you should add simply connected. Otherwise Enriques surfaces satisfy all the above, with k=9 $\endgroup$ Jan 20, 2019 at 11:01
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    $\begingroup$ @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. $\endgroup$
    – Tong
    Jan 20, 2019 at 11:20
  • $\begingroup$ 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 $\endgroup$ Jan 20, 2019 at 11:52
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    $\begingroup$ I am trying to understand what you said....are you saying the blow ups of fake projective spaces have Kahler-Einstein metric? $\endgroup$
    – Tong
    Jan 20, 2019 at 12:01
  • $\begingroup$ my mistake, no these are not, the canonical is not ample $\endgroup$ Jan 20, 2019 at 12:48

1 Answer 1


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

  • $\begingroup$ 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. $\endgroup$
    – Tong
    Jan 20, 2019 at 13:39

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