On a localization formula of epsilon factors via microlocal geometry

Abstract

Given a lisse $l$-adic sheaf $\mathcal{G}$ on a smooth proper variety $X$ and a lisse sheaf $\mathcal{ℱ}$ on an open dense $U$ in $X$, Kato and Saito conjectured a localization formula for the global $l$-adic epsilon factor ${\epsilon }_l\left(X,\mathcal{ℱ}\otimes \mathcal{G}\right)$ in terms of the global epsilon factor of $\mathcal{ℱ}$ and a certain intersection number associated to $det\left(\mathcal{G}\right)$ and the Swan class of $\mathcal{ℱ}$. In this article, we prove an analog of this conjecture for global de Rham epsilon factors in the classical setting of ${\mathcal{D}}_X$-modules on smooth projective varieties over a field of characteristic zero.