Chern Classes of Logarithmic Vector Fields for Locally Quasi-homogeneous Free Divisors

Let X be a nonsingular complex projective variety and D a locally quasihomogeneous 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 P. 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 CalderonMoreno, 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.