Chern classes of logarithmic vector fields
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 Csm(1U)=c(DerX(-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 Riemann-Roch type of formula is true if the Jacobian scheme of D is nonsingular or a complete intersection.