# 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 $h^{1,1} = h^{2,2} < h^{3,3} = h^{4,4} < h^{5,5} = h^{6,6} < \cdots h^{[\frac{n}{2}],[\frac{n}{2}]}$.

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

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$h^{1,1}=h^{2,2}<h^{3,3}=h^{4,4}<h^{5,5}=h^{6,6}<$ $h^{0,0}=h^{1,1}<h^{2,2}=h^{3,3}<h^{4,4}=h^{5,5}<h^{6,6}=\cdots.$ –  Ping May 23 '12 at 11:12
So strange! I don't why in the main text the two above formulae cannot be exhibited regularly and completely. So I give the formulae in the above comments. Sorry. –  Ping May 23 '12 at 11:15
Sorry. The correct two formulas is as follows. $$h^{1,1}=h^{2,2}<h^{3,3}=h^{4,4}<h^{5,5}=h^{6,6}<\cdots.$$ $$h^{0,0}=h^{1,1}<h^{2,2}=h^{3,3}<h^{4,4}=h^{5,5}<h^{6,6}=\cdots.$$ –  Ping May 23 '12 at 11:18