The construction problem for Hodge numbers modulo an integer

Abstract

For any integer $m\ge 2$ and any dimension $n\ge 1$, we show that any $n$-dimensional Hodge diamond with values in $\mathbb{Z}∕m\mathbb{Z}$ is attained by the Hodge numbers of an $n$-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of $n$-dimensional smooth complex projective varieties besides the ones induced by the Hodge symmetries, which answers a question raised by Kollár in 2012.