3 added 64 characters in body

Given any $n\geq 2$, is there an example of $n$-dimensional compact Kahler manifold such that its Hodge numbers satisfy $h^{1,1} = h^{2,2} < h^{3,3} = h^{4,4} < h^{5,5} = h^{6,6} < \cdots$.cdots h^{[\frac{n}{2}],[\frac{n}{2}]}$. Note that 2-plane complex Grassmannian's Hodge numbers satisfy$h^{0,0} = h^{1,1} < h^{2,2} = h^{3,3} < h^{4,4} = h^{5,5} < h^{6,6}=\cdots$h^{6,6}=\cdots h^{[\frac{n}{2}],[\frac{n}{2}]}$.

2 Fixed latex rendering problem

Given any $n\geq 2$, is there an example of $n$-dimensional compact Kahler manifold such that its Hodge numbers satisfy $h^{1,1}=h^{2,2} h^{1,1} = h^{2,2} < h^{3,3} = h^{4,4} < h^{5,5} = h^{6,6} < \cdots$.

Note that 2-plane complex Grassmannian's Hodge numbers satisfy $h^{0,0}=h^{1,1} h^{0,0} = h^{1,1} < h^{2,2} = h^{3,3} < h^{4,4} = h^{5,5} < h^{6,6}=\cdots$.

1

# the existence of compact Kahler manifolds satisfying some Hodge numbers' restrictions

Given any $n\geq 2$, is there an example of $n$-dimensional compact Kahler manifold such that its Hodge numbers satisfy $h^{1,1}=h^{2,2} Note that 2-plane complex Grassmannian's Hodge numbers satisfy$h^{0,0}=h^{1,1}