Filtration relative, l’idéal de Bernstein et ses pentes
Philippe Maisonobe · Rendiconti del Seminario Matematico della Università di Padova · 2023
Let f_i\colon X \rightarrow \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, \ldots, s_p] = \mathbf{C}_X [s_1, \ldots, 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, \ldots, f_p) the ideal of \mathbf{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 \mathcal{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 showed 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, \ldots, s_p) + \alpha_{H , i}) \in \mathcal{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 \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, \ldots, s_p] -modules related to our problem. This allows to specify the structure of the Bernstein’s ideals. Résumé Soit f_i \colon X \rightarrow {\mathbf C} , pour i entier compris entre 1 et p , des fonctions analytiques définies sur une variété analytique complexe X . Considérons {\cal D}_X l’anneau des opérateurs différentiels et {\cal D}_X[s_1, \ldots ,s_p]= {\mathbf C}_X[s_1, \ldots ,s_p] \otimes_{\mathbf C} {\cal D}_X . Soit m une section d’un {\cal D}_X -module holonome, notons {\cal B}(m,x_0, f_1, \ldots ,f_p) l’idéal de {\mathbf C}[s_1, \ldots ,s_p] des polynômes b vérifiant au voisinage de x_0 : 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} \; . C. Sabbah montre l’existence pour tout x_0 \in X d’un ensemble fini {\cal H} de formes linéaires à coefficients premiers entre eux dans {\mathbf N} telles que \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) \; ,