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

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.

References in corpus (5)

• Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities
• Grothendieck classes and Chern classes of hyperplane arrangements
• Normal crossing properties of complex hypersurfaces via logarithmic residues
• Characterizing normal crossing hypersurfaces
• Quasihomogeneous free divisors with only normal crossings in codimension one