Filtration Relative, l'Idéal de Bernstein et ses pentes

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