Hirzebruch-Milnor classes of local complete intersections, minimal exponent, and applications to higher singularities

TL;DR

This paper introduces spectral Hirzebruch-Milnor classes for local complete intersections, linking their vanishing to the minimal exponent and applying these classes to identify higher singularities like Du Bois and rational types.

Contribution

It develops a new framework for Hirzebruch-Milnor classes using deformation to the normal cone and Verdier-Saito specialization, connecting these classes to singularity invariants.

Findings

Vanishing of Hirzebruch-Milnor classes relates to the minimal exponent.

Hirzebruch-Milnor classes can detect higher Du Bois singularities.

Application to projective singular loci reveals higher rational singularities.

Abstract

In this paper we use the deformation to the normal cone and the corresponding Verdier-Saito specialization to define and study (spectral) Hirzebruch-Milnor type homology characteristic classes for local complete intersections. Our main results describe vanishing properties of these classes in relation to the minimal exponent. As applications, we show how Hirzebruch-Milnor classes of local complete intersections with a projective singular locus can be used to detect higher Du Bois and higher rational singularities.