Spectral Hirzebruch–Milnor classes of singular hypersurfaces

Abstract

We introduce spectral Hirzebruch–Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch–Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hyperplane section is not 1. These formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases. Here we use the Thom–Sebastiani theorem for the underlying filtered D-modules of vanishing cycles, from which we can deduce the Thom–Sebastiani type theorem for spectral Hirzebruch–Milnor classes. For the Chern classes after specializing to $$y=-\,1$$, we can give a relatively simple formula for the localized Milnor classes, which implies a new formula for the Euler numbers of projective hypersurfaces, using iterated hyperplane sections. Applications to log canonical thresholds and Du Bois singularities are also explained; for instance, the latter can be detected by using Hirzebruch–Milnor classes in the projective hypersurface case.