Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors

Let M be a complex manifold, D⊂ M a free divisor and U=M D its complement. In this paper we study the characteristic cycle CC(γ· U) of the restriction of a constructible function $γ$ 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 $γ$ and D. We prove that the log transversality condition is satisfied if either D is normal crossing and $γ$ is arbitrary, or D is holonomic strongly Euler homogheneous and $γ$ is non-characteristic. Under the log transversality assumption we establish a logarithmic pullback formula for CC(γ· U). Mixing Ginzburg's sharp construction with the logarithmic pullback, we obtain a double restriction formula for the Chern-Schwartz-MacPherson class c_*(γ· D∪ 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 Pⁿ D when D is a linear free divisor or a free hyperplane arrangement.