# The construction problem for Hodge numbers modulo an integer

## TL;DR

This paper proves that any Hodge diamond with values mod m can be realized by a smooth complex projective variety, showing the independence of Hodge numbers modulo m and answering a question about their relations.

## Contribution

It demonstrates that all possible Hodge diamonds modulo m are realizable, establishing the independence of Hodge numbers beyond known symmetries.

## Key findings

- Any n-dimensional Hodge diamond mod m is realizable by a smooth projective variety.
- No polynomial relations among Hodge numbers exist beyond Hodge symmetries.
- Answers Kollár's 2012 question on relations among Hodge numbers.

## Abstract

For any positive integer m and any dimension n, we show that any n-dimensional Hodge diamond with values in Z/mZ 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 Koll\'ar in 2012.