Comparison of the two notions of characteristic cycles

Abstract

Given a constructible sheaf $F$ on a complex manifold, Kashiwara-Schapira defined the notion of singular support and characteristic cycle of $F$. On the other hand for a Zariski constructible \'{e}tale sheaf $F$ on an algebraic variety $X$, Beilinson defined the notion of singular support of $F$ and Saito defined the notion of characteristic cycle of $F$. In this article we compare these notions and prove that they agree in a suitable sense. In the appendix we discuss extension of the notions of singular support and characteristic cycles developed by Umezaki-Yang-Zhao and Barrett.