Geometric Hodge structures with prescribed Hodge numbers

TL;DR

This paper demonstrates that any set of Hodge numbers satisfying basic constraints can be realized by a geometric Hodge structure, providing a simpler construction and exploring related 2D structures.

Contribution

It introduces a straightforward method to construct geometric Hodge structures with prescribed Hodge numbers, advancing understanding of their existence and properties.

Findings

Existence of geometric Hodge structures for arbitrary Hodge numbers

Simplified construction method compared to previous work

Discussion on 2-dimensional geometric Hodge structures

Abstract

Call a pure Hodge structure geometric if it is contained in the cohomology of a smooth complex projective variety. The main goal is to show that for any set of Hodge numbers (subject to the obvious constraints), there exists a geometric Hodge structure with precisely these Hodge numbers. This is related to recent work of Schreieder, but the construction here is simpler. This also contains some speculations about 2 dimensional geometric Hodge structures.