Motivic Chern classes of cones

TL;DR

This paper investigates motivic Chern classes of cones, revealing differences in various K-classes of projective cones of smooth curves and establishing connections between equivariant motivic Chern classes of projective varieties and their affine cones.

Contribution

It demonstrates the divergence of different K-classes for certain cones and generalizes results relating equivariant motivic Chern classes of projective and affine cones.

Findings

Different K-classes for cones of smooth curves

Connections between equivariant motivic Chern classes of projective and affine cones

Generalization of projective Thom polynomial results

Abstract

We study motivic Chern classes of cones. First we show examples of projective cones of smooth curves such that their various $K$-classes (sheaf theoretic, push-forward and motivic) are all different. Then we show connections between the torus equivariant motivic Chern class of a projective variety and of its affine cone, generalizing results on projective Thom polynomials.