Hodge symmetry and Lefschetz theorems for singular varieties

TL;DR

This paper explores the relationship between singularities in algebraic varieties and their Hodge-theoretic symmetries, establishing new results that connect singularity complexity with Hodge symmetry and Lefschetz properties.

Contribution

It introduces new theorems linking singularity conditions to Hodge symmetry and Lefschetz theorems, expanding understanding of Hodge theory in singular contexts.

Findings

Conditions on singularities relate to Hodge-Du Bois symmetry

Characterization of rational homology manifolds among low-dimensional varieties

New criteria for rational singularities in relation to Hodge structures

Abstract

We prove new results concerning the topology and Hodge theory of singular varieties. A common theme is that concrete conditions on the complexity of the singularities, from a number of different perspectives, are closely related to the symmetries of the Hodge-Du Bois diamond. We relate this to the theory of rational homology manifolds, and characterize these among low-dimensional varieties with rational singularities.