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Recently I am thinking about a problem. During this process I face an phenomenon, which is related to those Kahler manifolds whose Hodge numbers satisfy $h^{1,1}=h^{2,2}$. Except $CP^n$, are there any compact Kahler 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 hyperkahler manifolds, but all of them don't satisfy this condition.

Do some experts know any such an example except $CP^n$?

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 2012 at 11:45
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 Any smooth hypersurface of any degree in $\mathbb P^n$ for $n \geq 6$ has this property by the Lefschetz theorem. – Gunnar Magnusson May 13 2012 at 11:49
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 Cheap example: any threefold. – Artie Prendergast-Smith May 13 2012 at 14:22

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