Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities

TL;DR

This paper extends the Hirzebruch-Milnor class to hypersurfaces with nontrivial normal bundles, enabling detection of singularity types and spectral invariants, especially in projective cases with complex singularities.

Contribution

It introduces a method to define Hirzebruch-Milnor classes for hypersurfaces lacking global defining functions, linking these classes to singularity invariants and spectral data.

Findings

Hirzebruch-Milnor class can detect higher du~Bois and rational singularities.

Spectral Hirzebruch-Milnor class captures the minimal exponent of hypersurfaces.

Extension of classes is limited to projective hypersurfaces; non-projective cases remain challenging.

Abstract

We extend the Hirzebruch-Milnor class of a hypersurface $X$ to the case where the normal bundle is nontrivial and $X$ cannot be defined by a global function, using the associated line bundle and the graded quotients of the monodromy filtration. The earlier definition requiring a global defining function of $X$ can be applied rarely to projective hypersurfaces with non-isolated singularities. Indeed, it is surprisingly difficult to get a one-parameter smoothing with total space smooth without destroying the singularities by blowing-ups (except certain quite special cases). As an application, assuming the singular locus is a projective variety, we show that the minimal exponent of a hypersurface can be captured by the spectral Hirzebruch-Milnor class, and higher du~Bois and rational singularities of a hypersurface are detectable by the unnormalized Hirzebruch-Milnor class. Here the unnormalized class can be replaced by the normalized one in the higher du~Bois case, but for the higher rational case, we must use also the decomposition of the Hirzebruch-Milnor class by the action of the semisimple part of the monodromy (which is equivalent to the spectral Hirzebruch-Milnor class). We cannot extend these arguments to the non-projective compact case by Hironaka's example.