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.

To Appear in Mathematical Research Letters

References in corpus (3)

• Euler characteristics of general linear sections and polynomial Chern classes
• Towards transversality of singular varieties: splayed divisors
• Chern Classes of Logarithmic Vector Fields

Cited by in corpus (3)

• Euler characteristics of general linear sections and polynomial Chern classes
• Splayed divisors and their Chern classes
• Chern Classes of Logarithmic Derivations for Free Divisors with Jacobian Ideal of Linear Type