Chern Classes Of Logarithmic Vector Fields For Locally-Homogenous Free Divisors

Abstract

Let $X$ be a nonsingular complex projective variety and $D$ a locally
quasi-homogeneous free divisor in $X$. In this paper we study a numerical
relation between the Chern class of the sheaf of logarithmic derivations on $X$
with respect to $D$, and the Chern-Schwartz-MacPherson class of the complement
of $D$ in $X$. Our result confirms a conjectural formula for these classes, at
least after push-forward to projective space; it proves the full form of the
conjecture for locally quasi-homogeneous free divisors in $\mathbb P^n$. The
result generalizes several previously known results. For example, it recovers a
formula of M. Mustata and H. Schenck for Chern classes for free hyperplane
arrangements. Our main tools are Riemann-Roch and the logarithmic comparison
theorem of Calderon-Moreno, Castro-Jimenez, Narvaez-Macarro, and David Mond. As
a subproduct of the main argument, we also obtain a schematic Bertini statement
for locally quasi-homogeneous divisors.