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An introduction to $V$-filtrations

Qianyu Chen, Bradley Dirks, Mircea Mustaţă

Abstract

We give an introduction to the theory of $V$-filtrations of Malgrange and Kashiwara. After discussing the basic properties of this construction (in the case of a smooth hypersurface and, later, in the general case), we describe the connection with the theory of $b$-functions. As an example, we treat the case of weighted homogeneous isolated singularities. We discuss the compatibility of $V$-filtrations with proper push-forward and duality and the connection with nearby and vanishing cycles via the Riemann-Hilbert correspondence. We end by describing some invariants of singularities via the $V$-filtration.

An introduction to $V$-filtrations

Abstract

We give an introduction to the theory of -filtrations of Malgrange and Kashiwara. After discussing the basic properties of this construction (in the case of a smooth hypersurface and, later, in the general case), we describe the connection with the theory of -functions. As an example, we treat the case of weighted homogeneous isolated singularities. We discuss the compatibility of -filtrations with proper push-forward and duality and the connection with nearby and vanishing cycles via the Riemann-Hilbert correspondence. We end by describing some invariants of singularities via the -filtration.
Paper Structure (30 sections, 46 theorems, 357 equations)

This paper contains 30 sections, 46 theorems, 357 equations.

Table of Contents

  1. Introduction
  2. A brief review of $\mathcal{D}$-module theory
  3. The sheaf of differential operators
  4. The de Rham complex
  5. The left-right equivalence
  6. Singular support and holonomic $\mathcal{D}$-modules
  7. Functors on $\mathcal{D}$-modules
  8. The Riemann-Hilbert correspondence
  9. The $V$-filtration with respect to a smooth hypersurface
  10. The filtration $V^{\bullet}\mathcal{D}_X$
  11. The $V$-filtration with respect to $t$
  12. The maps ${\rm Var}$ and ${\rm can}$
  13. The $V$-filtration with respect to an arbitrary hypersurface
  14. $V$-filtrations and $b$-functions
  15. Basics on $b$-functions
  16. ...and 15 more sections

Key Result

Lemma 3.2

The operator $s=-\partial_tt$ satisfies the following properties:

Theorems & Definitions (144)

  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Remark 3.6: Restriction of $V$-filtration to an open subset
  • Remark 3.7: Behavior on $X\smallsetminus H$
  • Remark 3.8: Independence of coordinates and of the equation $t$
  • Remark 3.9
  • ...and 134 more