Filtration Relative, l'Idéal de Bernstein et ses pentes

Abstract

Let $f_i:X\to \mathbf{C}$, for $i$ integer between $1$ and $p$, be analytic functions defined on a complex analytic variety $X$. Let us consider ${\mathcal{D}}_X$ the ring of linear differential operators and ${\mathcal{D}}_X[s_1,\dots ,s_p]={\mathbf{C}}_X[s_1,\dots ,s_p]{\otimes }_{\mathbf{C}}{\mathcal{D}}_X$. Let $m$ be a section of a holonomic ${\mathcal{D}}_X$-Module. We denote $\mathcal{B}(m,x_0,f_1,\dots ,f_p)$ the ideal of $\mathbf{C}[s_1,\dots ,s_p]$ constituted by the polynomials $b$ satisfying in the neighborhood of $x_0\in X$ : $$B(s_1,\dots ,s_p)m{f}_1^{s_1}\dots f_p^{s_p}\in {\mathcal{D}}_X[s_1,\dots ,s_p]m{f}_1^{s_1+1}\dots f_p^{s_p+1}.$$ This ideal is called Bernstein's ideal. C. Sabbah shows the existence for every $x_0\in X$ of a finite set $\mathcal{H}$ of linear forms with coefficients in $\mathbf{N}$, such that: $$\prod _{H\in \mathcal{H}}\prod _{i\in I_{\mathcal{H}}}(H(s_1,\dots ,s_p)+{\alpha }_{H,i})\in \mathcal{B}(m,x_0,f_1,\dots ,f_p),$$ where ${\alpha }_{H,i}$ are complex numbers. The purpose of this article is to show in particular the existence of a minimal set $\mathcal{H}$. In addition, when $m$ is a section of a holonomic regular ${\mathcal{D}}_X$-Module, we will precise geometrically this set from the characteristic variety of ${\mathcal{D}}_X$-Module generated by $m$. We introduce and study especially the relative characteristic variety of the ${\mathcal{D}}_X[s_1,\dots ,s_p]$ - Modules related to our problem. This allows to specify the structure of the Bernstein's ideals.

in French