The Irregularity of the Direct Image of Some $\mathcal{D}$-modules

Abstract

Let $f$ and $g$ be two regular functions on $U$ smooth affne variety. Let $\mathcal{M}$ be a regular holonomic ${\mathcal{D}}_U$-module. We are interested in the irregularity of the complex $f_{+}(\mathcal{M}{e}^g)$. More precisely, we relate the irregularity number at $c$ of the systems ${\mathcal{H}}^k{f}_{+}(\mathcal{M}{e}^g)$ with the characteristic cycles of the systems ${\mathcal{H}}^k(f,g{)}_{+}(\mathcal{M})$.