K-theoretic defect in Chern class identity for a free divisor

TL;DR

This paper investigates the relationship between motivic Chern classes and sheaves of logarithmic differentials for free divisors, providing explicit calculations in specific geometric contexts.

Contribution

It introduces a K-theoretic defect in the Chern class identity for free divisors and computes this difference explicitly in certain cases.

Findings

Explicit difference formulas for free divisors on surfaces

Calculations for hyperplane arrangements with free affine cones

Identification of a K-theoretic defect in Chern class identities

Abstract

Let $X$ be a nonsingular variety defined over an algebraically closed field of characteristic $0$, and $D$ be a free divisor. We study the motivic Chern class of $D$ in the Grothendieck group of coherent sheaves $G_0(X)$, and another class defined by the sheaf of logarithmic differentials along $D$. We give explicit calculations of the difference of these two classes when: $D$ is a divisor on a nonsingular surface; $D$ is a hyperplane arrangement whose affine cone is free.