On the construction problem for Hodge numbers

Abstract

For any symmetric collection ${\left(h^{p,q}\right)}_{p+q=k}$ of natural numbers, we construct a smooth complex projective variety $X$ whose weight-$k$ Hodge structure has Hodge numbers $h^{p,q}\left(X\right)=h^{p,q}$; if $k=2m$ is even, then we have to impose that $h^{m,m}$ is bigger than some quadratic bound in $m$. Combining these results for different weights, we solve the construction problem for the truncated Hodge diamond under two additional assumptions. Our results lead to a complete classification of all nontrivial dominations among Hodge numbers of Kähler manifolds.