Hodge-de Rham numbers of almost complex 4-manifolds

TL;DR

This paper introduces Hodge-de Rham numbers for almost complex 4-manifolds, extending classical invariants of complex surfaces, and demonstrates their role in obstructing complex structures without relying on surface classification.

Contribution

It generalizes Hodge numbers to almost complex 4-manifolds and explores their properties, showing they are mostly determined by cohomology and can obstruct complex structures.

Findings

Hodge-de Rham numbers are determined by cohomology except for irregularity.

These numbers extend classical Hodge invariants to a broader setting.

They can be used to rule out complex structures on certain manifolds.

Abstract

We introduce and study Hodge-de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general setting, and it is shown that all Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity). Finally, these numbers are shown to prohibit the existence of complex structures on certain manifolds, without reference to the classification of surfaces.