Q-factoriality and Hodge-Du Bois theory

TL;DR

This paper establishes Hodge-theoretic formulas to measure the $Q$-factoriality defect of normal projective varieties and analytic germs, leading to various geometric and topological consequences.

Contribution

It introduces new formulas linking Hodge theory to $Q$-factoriality defects, with applications to Samuel's conjecture, rational homology threefolds, and invariance under flops.

Findings

Formulas for $Q$-factoriality defect using Hodge theory

Characterization of rational homology threefolds with rational singularities

Invariance of Hodge-Du Bois numbers under flops

Abstract

We prove Hodge-theoretic formulas for the $\mathbb{Q}$-factoriality defect of a normal projective variety, and for the local analytic $\mathbb{Q}$-factoriality defect of an analytic germ of a normal variety. These formulas lead to consequences ranging from a local analytic version of Samuel's conjecture to a characterization of projective rational homology threefolds with rational singularities, or the invariance of Hodge-Du Bois numbers under flops of projective threefolds.