On the construction problem for Hodge numbers

TL;DR

This paper addresses the construction problem for Hodge numbers, providing methods to realize specified symmetric collections as Hodge numbers of smooth complex projective varieties, and classifies domination relations among Hodge numbers of Kähler manifolds.

Contribution

It introduces a construction technique for realizing prescribed Hodge numbers and offers a complete classification of domination relations among Hodge numbers of Kähler manifolds.

Findings

Constructed varieties with prescribed Hodge numbers for various weights.

Established bounds on diagonal Hodge numbers for even weights.

Classified all nontrivial domination relations among Hodge numbers.

Abstract

For any symmetric collection of natural numbers h^{p,q} with p+q=k, we construct a smooth complex projective variety whose weight k Hodge structure has these Hodge numbers; 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 Kaehler manifolds.