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Jul 19, 2016 at 21:26 history edited Francesco Polizzi CC BY-SA 3.0
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Jul 19, 2016 at 13:51 vote accept Vamsi
Jul 19, 2016 at 13:51 comment added abx In fact, for a surface with $c_1^2=3c_2$, an easy computation gives $h^{1,1}=p_g+q+1$. So the answer is no for any such surface which is not a fake projective plane.
Jul 19, 2016 at 13:44 history answered Francesco Polizzi CC BY-SA 3.0