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Recently I have been thinking about a problem. During this process I faced a phenomenon which is related to those Kähler manifolds whose Hodge numbers satisfy $h^{1,1}=h^{2,2}$. Except $\mathbf{CP}^n$, are there any compact Kähler manifolds whose Hodge numbers satisfy $h^{1,1}=h^{2,2}$? At this moment, I don't have any such examples. I have calculated $A_n$-type flag manifolds and some hyper-Kähler manifolds, but none of them satisfy this condition.

Do some experts know any such an example other than $\mathbf{CP}^n$?

Thanks in advance!

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Sorry. I suddenly realize that my favoriate manifolds $CP^n\sharp\bar{CP^n}$ ($n\geq 2$) satisfy this condition:-) Are there any more such examples? –  Ping May 13 '12 at 11:45
Any smooth hypersurface of any degree in $\mathbb P^n$ for $n \geq 6$ has this property by the Lefschetz theorem. –  Gunnar Þór Magnússon May 13 '12 at 11:49
Cheap example: any threefold. –  Artie Prendergast-Smith May 13 '12 at 14:22
Also, fake projective planes: en.wikipedia.org/wiki/Fake_projective_plane –  Daniel Litt Jul 25 '13 at 8:23

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