Motivic Milnor classes

TL;DR

This paper introduces a motivic framework for Milnor classes, generalizing classical invariants and establishing new natural transformations and formulas within algebraic geometry.

Contribution

It develops a motivic Grothendieck group and natural transformations capturing Milnor and Hirzebruch--Milnor classes, along with a Verdier-type Riemann--Roch formula.

Findings

Defined a motivic Grothendieck group for local complete intersections.

Expressed Milnor and Hirzebruch--Milnor classes as special values of natural transformations.

Established a Verdier-type Riemann--Roch formula for the motivic Hirzebruch-Milnor class.

Abstract

The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern--Schwartz--MacPherson's class and Fulton--Johnson's canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a "motivic" Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch--Milnor class, as a special value of a distinguished element under these natural transformations. We also show a Verdier-type Riemann--Roch formula for our motivic Hirzebruch-Milnor class. We use Fulton--MacPherson's bivariant theory and the motivic Hirzebruch class.