Higher Du Bois and higher rational singularities of hypersurfaces

TL;DR

This paper characterizes higher Du Bois and higher rational singularities of hypersurfaces using Hodge theory and spectral Hirzebruch-Milnor classes, providing new criteria for their detection.

Contribution

It offers several equivalent characterizations of these singularities and introduces a homological criterion based on spectral classes, advancing understanding of hypersurface singularities.

Findings

Characterizations via Hodge filtration on vanishing cycle modules

Homological criterion using spectral Hirzebruch-Milnor classes

Connections between singularities and Hodge-theoretic invariants

Abstract

In this note, we give several equivalent characterizations of higher Du Bois and higher rational singularities in the context of globally defined hypersurfaces. As a key input, we characterize these singularities using the Hodge filtration on the vanishing cycle mixed Hodge module. As an application, we indicate a homological criterion for detecting higher Du Bois and higher rational hypersurface singularities, formulated in terms of the spectral Hirzebruch-Milnor characteristic classes.