Chern classes of logarithmic vector fields for locally quasi-homogeneous free divisors

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 pushforward 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. Mustaţă 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.

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Received 25 October 2012

Accepted 25 October 2012

Published 3 August 2018