Éventails associés à des fonctions analytiques

Let $X$ be a complex analytic manifold, $(f_1,\dots, f_p)$ be analytic functions on $X$, and denote by $F=f_1\dots f_p$ their product. Given a regular holonomic $\mathcal D_X$-module $\mathcal M$ and a section $m\in\mathcal M$, one can associate to the characteristic variety of the $\mathcal D_X[s_1,\ldots,s_p]$-module $\mathcal D_X[s_1,\ldots ,s_p]m f_1^{s_1}\dots f_p^{s_p}$ a finite set $\mathcal H_{f,m}$ of hyperplanes in $\mathbf C^p$. We study this characteristic variety and prove that the set $\mathcal H_{f,m}$ is contained in the union of the coordinate hyperplanes of $\mathbf C^p$ if and only if the morphism $f:\mathbf C^n \rightarrow \mathbf C^p$ has no blowing up in codimension zero and its critical locus is contained in the set $F=0$.