Chern Classes of Logarithmic Vector Fields

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 $c_{SM}(1_U)=c({\text{Der}}_X(-\log D))\cap [X]$ 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.

Cited by in corpus (2)

• Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors
• Chern Classes of Logarithmic Derivations for Free Divisors with Jacobian Ideal of Linear Type