Filtered restriction of geometric D-modules

TL;DR

This paper advances the computation of restrictions of D-modules by incorporating order filtrations, enabling minimal presentations of local cohomology modules with pole order filtrations for quasi-homogeneous polynomials with isolated singularities.

Contribution

It introduces a new approach to compute restrictions of D-modules that respects order filtrations, improving the understanding of local cohomology modules with pole order filtrations.

Findings

Respects the order filtration in restriction process

Computes minimal presentation of algebraic local cohomology modules

Applies to quasi-homogeneous polynomials with isolated singularities

Abstract

In D-module theory, we have the notion of the restriction of a module along a smooth variety. T. Oaku and N. Takayama have described a process to compute the restriction, which starts from a free resolution adapted to the V-filtration of Kashiwara-Malgrange. By using the theory of free resolutions adapted to the (order and V)-bifiltration (from T. Oaku , N. Takayama and M. Granger), we endow the restriction with an order filtration and we show that the process of Oaku and Takayama respects the order filtration. In consequence, we compute a minimal presentation of the algebraic local cohomology module associated with a quasi-homogeneous polynomial with an isolated singularity, viewed as a D-module and endowed with a kind of a pole order filtration.