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Three results on holonomic D-modules

Claude Sabbah

Abstract

In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham complex after localization and dual localization of a holonomic D-module along a hypersurface, as well as after tensoring with a rank one meromorphic connection with regular singularities. II. (Local generic vanishing theorems for holonomic D-modules) We prove that the natural morphism from the proper pushforward to the total pushforward of an algebraic holonomic D-module by an open inclusion is an isomorphism if we first twist the D-module structure by suitable closed algebraic differential forms. III. (Laplace transform of a Stokes-filtered constructible sheaf of exponential type) Motivated by the construction in [YZ24], we~propose a slightly different construction of the Laplace transform of a Stokes-perverse sheaf on the projective line and show directly that it corresponds to the Laplace transform of the corresponding holonomic D-module via the Riemann-Hilbert-Birkhoff-Deligne-Malgrange correspondence. This completes the presentation given in [Sab13, Chap. 7]}, where only the other direction of the Laplace transformation is analyzed. We~also compare our approach with the construction made previously in [YZ24].

Three results on holonomic D-modules

Abstract

In this text, we illustrate the use of local methods in the theory of (irregular) holonomic D-modules. I. (The Euler characteristic of the de~Rham complex) We show the invariance of the global or local Euler characteristic of the de~Rham complex after localization and dual localization of a holonomic D-module along a hypersurface, as well as after tensoring with a rank one meromorphic connection with regular singularities. II. (Local generic vanishing theorems for holonomic D-modules) We prove that the natural morphism from the proper pushforward to the total pushforward of an algebraic holonomic D-module by an open inclusion is an isomorphism if we first twist the D-module structure by suitable closed algebraic differential forms. III. (Laplace transform of a Stokes-filtered constructible sheaf of exponential type) Motivated by the construction in [YZ24], we~propose a slightly different construction of the Laplace transform of a Stokes-perverse sheaf on the projective line and show directly that it corresponds to the Laplace transform of the corresponding holonomic D-module via the Riemann-Hilbert-Birkhoff-Deligne-Malgrange correspondence. This completes the presentation given in [Sab13, Chap. 7]}, where only the other direction of the Laplace transformation is analyzed. We~also compare our approach with the construction made previously in [YZ24].
Paper Structure (38 sections, 8 theorems, 104 equations)

This paper contains 38 sections, 8 theorems, 104 equations.

Table of Contents

  1. The Euler characteristic of the de Rham complex
  2. Introduction to Part \ref{['part:Euler']}
  3. Proof of Theorem \ref{['th:Eulerlocalgood']}
  4. Setting
  5. A reminder of the notion of good formal structure
  6. Omitting the Stokes filtration
  7. Reduction to the case "good of rank one"
  8. Computation of the irregularity number of a good meromorphic flat bundle of rank one
  9. Reduction of the proof of Theorem \ref{['th:Eulerlocal']} to that of Theorem \ref{['th:Eulerlocalgood']}
  10. Local generic vanishing theorems for holonomic D-modules
  11. Local and relative generic vanishing theorems
  12. A general result
  13. Applications
  14. A criterion for local vanishing
  15. Reduction of Proposition \ref{['prop:criterion']} to the elementary case
  16. ...and 23 more sections

Key Result

proposition 1

Assume that the transverse monodromy of the formal regular part of $\cM$ along the smooth part of $D$ has no eigenvalue which is a root of unity of order $\leq d$ ( if this formal regular part is zero). Then the natural morphism $\cM(!D)\to\cM=\cM(*D)$ is an isomorphism.

Theorems & Definitions (38)

  • proof : Proof that Theorem \ref{['th:Eulerlocal']} implies Theorem \ref{['th:Euler']}
  • definition 1: Irregularity number
  • proof
  • definition 2: Relative vanishing
  • proof
  • definition 3: Non-resonance
  • definition 4: Bibi93 and Mochizuki08
  • proof
  • proof
  • proposition 1
  • ...and 28 more