Higher Du Bois and higher rational singularities of hypersurfaces

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.