Spectral Hirzebruch-Milnor classes of singular hypersurfaces

TL;DR

This paper introduces spectral Hirzebruch-Milnor classes for singular hypersurfaces, linking local spectral data with global invariants, and provides formulas and applications for calculating invariants like Euler numbers and detecting singularity types.

Contribution

It defines spectral Hirzebruch-Milnor classes using vanishing cycles and Todd classes, and establishes formulas and theorems for their computation and applications.

Findings

Spectral Hirzebruch-Milnor classes coincide with Steenbrink spectra in isolated singularities.

Formulas for Hirzebruch-Milnor classes in terms of hyperplane sections are derived.

New formulas for Euler numbers and detection of Du Bois singularities are provided.

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.