Gr\"obner basics for mixed Hodge modules

TL;DR

This paper develops a Gr"obner basis theory for a class of algebras that generalizes PBW-algebras and differential operator rings, focusing on computing filtrations relevant to mixed Hodge modules.

Contribution

It introduces a new Gr"obner basis framework tailored for bifiltered D-modules associated with mixed Hodge modules, enabling effective filtration computations.

Findings

Provides methods to compute filtrations and graded structures.

Applies to bifiltered D-modules satisfying mixed Hodge module properties.

Facilitates computation of the order filtration on V-filtration pieces.

Abstract

We develop a Gr\"obner basis theory for a class of algebras that generalizes both PBW-algebras and rings of differential algebras on smooth varieties. Emphasis lies on methods to compute filtrations and graded structures defined by weight vectors. The approach is tailored for bifiltered D-modules satisfying properties of mixed Hodge modules. As a key ingredient in functors of such modules our theory applies to compute the order filtration on pieces of a V-filtration.