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!