Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors

TL;DR

This paper extends Ginzburg's sharp operation to logarithmic versions for free divisors, establishing new transversality conditions and formulas for characteristic cycles and Chern-Schwartz-MacPherson classes, with applications to Euler characteristics.

Contribution

It introduces the log transversality condition and a logarithmic pullback formula, broadening the toolkit for analyzing characteristic cycles and CSM classes in free divisor settings.

Findings

Log transversality condition holds for normal crossing and certain holonomic divisors.

Logarithmic pullback formula for characteristic cycles is established.

Applications include non-negativity of Euler characteristics and CSM classes for specific hypersurfaces.

Abstract

Let $M$ be a complex manifold, $D\subset M$ a free divisor and $U=M\setminus D$ its complement. In this paper we study the characteristic cycle $\textup{CC}(\gamma\cdot \ind_U)$ of the restriction of a constructible function $\gamma$ on $U$. We globalise Ginzburg's local sharp construction and introduce the log transversality condition, which is a new transversality condition about the relative position of $\gamma$ and $D$. We prove that the log transversality condition is satisfied if either $D$ is normal crossing and $\gamma$ is arbitrary, or $D$ is holonomic strongly Euler homogheneous and $\gamma$ is non-characteristic. Under the log transversality assumption we establish a logarithmic pullback formula for $\textup{CC}(\gamma\cdot \ind_U)$. Mixing Ginzburg's sharp construction with the logarithmic pullback, we obtain a double restriction formula for the Chern-Schwartz-MacPherson class $c_*(\gamma\cdot \ind_{D\cup V})$ where $V$ is any reduced hypersurface in $M$. Applications of our results include the non-negativity of Euler characteristics of effective constructible functions, and CSM classes of hypersurfaces in the open manifold $\mathbb{P}^n\setminus D$ when $D$ is a linear free divisor or a free hyperplane arrangement.