Chern Classes of Logarithmic Vector Fields

TL;DR

This paper investigates the conditions under which a specific Chern class formula for logarithmic vector fields holds on complex varieties, linking it to Riemann-Roch formulas and singularity invariants.

Contribution

It establishes the equivalence of the Chern class formula with a Riemann-Roch type formula and characterizes its validity in terms of singularity invariants and geometric conditions.

Findings

The formula is equivalent to a Riemann-Roch type formula.

On surfaces, the formula holds iff Milnor number equals Tjurina number.

The formula is valid if the Jacobian scheme is nonsingular or a complete intersection.

Abstract

Let $X$ be a nonsingular complex variety and $D$ a reduced effective divisor in $X$. In this paper we study the conditions under which the formula $c_{SM}(1_U)=c(\textup{Der}_X(-\log D))\cap [X]$ is true. We prove that this formula is equivalent to a Riemann-Roch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of $D$. We also show the Rimann-Roch type of formula is true if the Jacobian scheme of $D$ is nonsingular or a complete intersection.