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On the D-module of an isolated singularity

Thomas Bitoun

TL;DR

This work addresses describing the D-module generated by powers of $\frac{1}{f}$ for an isolated hypersurface singularity $Z=\{f=0\}$ in dimension $n\ge 3$ through the pole order filtration on the de Rham cohomology $H'$ of the complement of $Z$. It introduces a residue-based morphism $r$ from $\mathcal{O}(\star Z)_o$ to $\delta_o \otimes H'$ whose kernel is the intersection cohomology D-module $\mathcal{L}_o$, providing a new lens on Vilonen's theorem and enabling analysis of D-submodules generated by $\frac{1}{f^l}$ and by Hodge/pole-filtered pieces. The paper derives explicit length formulas: the length of $D_o \frac{1}{f^{l+1}} / \mathcal{L}_o$ equals $\dim P_l H'$, with equality in the quasi-homogeneous case, and extends these ideas to algebraic D-modules, linking lengths to pole-order filtrations on the de Rham cohomology of hypersurface complements. Overall, the approach connects D-module theory, Hodge theory, and singularity theory to yield computable invariants and a clearer picture of how pole order and Hodge filtrations govern module structure.

Abstract

Let Z be the germ of a complex hypersurface isolated singularity of equation f, with Z at least of dimension 2. We consider the family of analytic D-modules generated by the powers of 1/f and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of {f=0} in the neighborhood of the singularity.

On the D-module of an isolated singularity

TL;DR

This work addresses describing the D-module generated by powers of for an isolated hypersurface singularity in dimension through the pole order filtration on the de Rham cohomology of the complement of . It introduces a residue-based morphism from to whose kernel is the intersection cohomology D-module , providing a new lens on Vilonen's theorem and enabling analysis of D-submodules generated by and by Hodge/pole-filtered pieces. The paper derives explicit length formulas: the length of equals , with equality in the quasi-homogeneous case, and extends these ideas to algebraic D-modules, linking lengths to pole-order filtrations on the de Rham cohomology of hypersurface complements. Overall, the approach connects D-module theory, Hodge theory, and singularity theory to yield computable invariants and a clearer picture of how pole order and Hodge filtrations govern module structure.

Abstract

Let Z be the germ of a complex hypersurface isolated singularity of equation f, with Z at least of dimension 2. We consider the family of analytic D-modules generated by the powers of 1/f and describe it in terms of the pole order filtration on the de Rham cohomology of the complement of {f=0} in the neighborhood of the singularity.
Paper Structure (5 sections, 9 theorems, 9 equations)

This paper contains 5 sections, 9 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. A morphism of $\mathcal{D}$-modules
  3. Setup and conventions
  4. Construction
  5. Some corollaries

Key Result

Lemma 2.2.1

Let $V$ be a finite-dimensional complex vector space and let $o$ be a point of $X.$ For $\Omega_o^n$ the stalk at $o$ of the sheaf of differential $n$-forms, the space of linear maps $L(\Omega_o^n, V)$ from the right $D_o$-module $\Omega_o^n$ to $V$ is naturally a $D_o$-module. Moreover the $D_o$-su

Theorems & Definitions (22)

  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2
  • proof
  • Theorem 2.2.3: Vilonen
  • proof
  • Theorem 2.2.4
  • proof
  • Remark 2.2.5
  • Corollary 2.2.6
  • ...and 12 more