EDIT: this answer refers to a previous version of the question.
Already for $n=3$ the answer is no. Indeed, $h^{3,3}=1$ so by your condition $h^{1,1}=h^{2,2}=0$ but a compact Kähler manifold has $h^{1,1}>0$.
|
2 | added 69 characters in body | ||
|
|
||||
|
|
1 |
|
||
|
Already for $n=3$ the answer is no. Indeed, $h^{3,3}=1$ so by your condition $h^{1,1}=h^{2,2}=0$ but a compact Kähler manifold has $h^{1,1}>0$. |
||||
