[Paper Review] Survey on the $D$-module $f^s$

Anton Leykin, Uli Walther|arXiv (Cornell University)|Apr 28, 2015

Algebraic Geometry and Number Theory115 references3 citations

TL;DR

This survey provides a comprehensive overview of the $D$-module $\mathscr{M}_f(s) = D[s]/\operatorname{ann}_{D[s]}(f^s)$, focusing on its role in singularity theory through the Bernstein–Sato polynomial, multiplier ideals, and monodromy. It details algorithmic methods for computing these invariants, especially for hyperplane arrangements and free divisors, using Gröbner bases and elimination techniques in the Weyl algebra.

ABSTRACT

In this survey we discuss various aspects of the singularity invariants with differential origin derived from the $D$-module generated by $f^s$.

Motivation & Objective

• To systematize and survey the theory of the $D$-module $\mathscr{M}_f(s)$ generated by $f^s$, a central object in algebraic $D$-module theory.
• To explain how invariants like the Bernstein–Sato polynomial, multiplier ideals, and monodromy of the Milnor fiber arise from $\mathscr{M}_f(s)$.
• To present algorithmic methods for computing these invariants, especially in the context of hyperplane arrangements and free divisors.
• To connect the $D$-module structure to key singularity invariants such as the logarithmic comparison theorem and $V$-filtration.

Proposed method

• Utilizes the parametric annihilator $\operatorname{ann}_{D[s]}(f^s)$ to define the cyclic $D[s]$-module $\mathscr{M}_f(s)$, which encodes the functional equation $P \bullet f^{s+1} = b_f(s) f^s$.
• Employs Gröbner basis techniques in the Weyl algebra $D[s]$ to compute the Bernstein–Sato polynomial $b_f(s)$ as the minimal polynomial of $s$ modulo the annihilator.
• Applies elimination theory via the weight $(-w,w)$ on $D_{x,t}[s]$ to compute $\operatorname{in}_{(-w,w)}(I_f) \cap \mathbb{C}[x,s]$, which captures local Bernstein–Sato polynomials.
• Uses the generalized functional equation $b(\sigma)gf^s = \sum P_k g f_k f^s$ with $\sigma = -\sum \partial_{t_i} t_i$ to compute multiplier ideals and jumping numbers.
• Leverages the $V$-filtration and characteristic cycle to analyze the singular support and holonomicity of $\mathscr{M}_f(s)$.
• Integrates computational tools from Singular, Macaulay2, and Risa/Asir via specialized libraries (e.g., dmod_lib, D-modules package) for practical implementation.

Experimental results

Research questions

• RQ1 How can the Bernstein–Sato polynomial $b_f(s)$ be algorithmically computed from the $D$-module $\mathscr{M}_f(s)$ using Gröbner bases and elimination?
• RQ2 What is the relationship between the $V$-filtration on $\mathscr{M}_f(s)$ and the roots of the Bernstein–Sato polynomial?
• RQ3 How do multiplier ideals and jumping numbers arise from the generalized functional equation in the multi-variable setting?
• RQ4 What stratifications of $\mathbb{C}^n$ are induced by local Bernstein–Sato polynomials, and how can they be computed algorithmically?
• RQ5 In what cases (e.g., hyperplane arrangements, free divisors) does the logarithmic comparison theorem hold, and how does it relate to $\mathscr{M}_f(s)$?

Key findings

• The Bernstein–Sato polynomial $b_f(s)$ is the monic generator of the ideal in $\mathbb{C}[s]$ generated by all $b_{f,P}(s)$ from functional equations $P \bullet f^{s+1} = b_{f,P}(s) f^s$.
• The $D[s]$-module $\mathscr{M}_f(s)$ is holonomic and coherent, with characteristic variety of dimension $n$, and its characteristic cycle encodes multiplicities of components in the graded object.
• For hyperplane arrangements, the Bernstein–Sato polynomial can be computed algorithmically via Gröbner deformation and elimination, as shown in [5] and [158].
• The generalized Bernstein–Sato polynomial $b_{f,g}(\sigma)$ for ideals allows computation of jumping numbers and multiplier ideals, with implementations available in Macaulay2 via the MultiplierIdeals package.
• The $V$-filtration on $\mathscr{M}_f(s)$ provides a filtration whose associated graded pieces control the roots of $b_f(s)$, linking the module to the singularity structure.
• Algorithms for computing $b_f(s)$ and multiplier ideals are implemented in software such as Singular (dmod_lib), Macaulay2 (D-modules package), and Risa/Asir, with performance comparisons in [129].