Article

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

05/2012;
Source: arXiv

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.

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### Keywords

Castro-Jimenez

Chern class

Chern classes

Chern-Schwartz-MacPherson class

classes

conjectural formula

conjecture

David Mond

H. Schenck

known results

logarithmic derivations

main argument

main tools

nonsingular complex projective variety

schematic Bertini statement

sheaf