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Kashiwara conjugation and the enhanced Riemann-Hilbert correspondence

Andreas Hohl

TL;DR

This work extends Kashiwara’s conjugation to the irregular Riemann–Hilbert setting by proving compatibility of the Kashiwara conjugation with the enhanced De Rham functor $\,\mathcal{DR}^E_X$, leveraging Hermitian duality and Sabbah–Kedlaya–Mochizuki’s holonomic theory. It then develops Galois descent for enhanced ind-sheaves beyond compactness, showing that $\,\mathbb{R}$-constructible enhanced ind-sheaves with a Galois action descend from extensions of scalars. A central theme is the transfer of $K$-structures on enhanced solution data to the level of monodromy, showing that Stokes and generalized monodromy matrices can be defined over the subfield $K\subset\mathbb{C}$ when a $K$-structure exists. The paper connects meromorphic normal forms, their descent to $K$-lattices, and the resulting implications for Stokes data, thus providing a unified, intrinsic pathway from algebraic descent to topological monodromy data in the irregular Riemann–Hilbert framework.

Abstract

We study some aspects of conjugation and descent in the context of the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara. First, we give a proof of the fact that Kashiwara's conjugation functor for holonomic D-modules is compatible with the enhanced De Rham functor. Afterwards, we work out some complements on Galois descent for enhanced ind-sheaves, slightly generalizing results obtained in previous joint work with Barco, Hien and Sevenheck. Finally, we show how local decompositions of an enhanced ind-sheaf into exponentials descend to lattices over smaller fields. This shows in particular that a structure of the enhanced solutions of a meromorphic connection over a subfield of the complex numbers has implications on its generalized monodromy data (in particular, the Stokes matrices), generalizing and simplifying an argument given in our previous work.

Kashiwara conjugation and the enhanced Riemann-Hilbert correspondence

TL;DR

This work extends Kashiwara’s conjugation to the irregular Riemann–Hilbert setting by proving compatibility of the Kashiwara conjugation with the enhanced De Rham functor , leveraging Hermitian duality and Sabbah–Kedlaya–Mochizuki’s holonomic theory. It then develops Galois descent for enhanced ind-sheaves beyond compactness, showing that -constructible enhanced ind-sheaves with a Galois action descend from extensions of scalars. A central theme is the transfer of -structures on enhanced solution data to the level of monodromy, showing that Stokes and generalized monodromy matrices can be defined over the subfield when a -structure exists. The paper connects meromorphic normal forms, their descent to -lattices, and the resulting implications for Stokes data, thus providing a unified, intrinsic pathway from algebraic descent to topological monodromy data in the irregular Riemann–Hilbert framework.

Abstract

We study some aspects of conjugation and descent in the context of the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara. First, we give a proof of the fact that Kashiwara's conjugation functor for holonomic D-modules is compatible with the enhanced De Rham functor. Afterwards, we work out some complements on Galois descent for enhanced ind-sheaves, slightly generalizing results obtained in previous joint work with Barco, Hien and Sevenheck. Finally, we show how local decompositions of an enhanced ind-sheaf into exponentials descend to lattices over smaller fields. This shows in particular that a structure of the enhanced solutions of a meromorphic connection over a subfield of the complex numbers has implications on its generalized monodromy data (in particular, the Stokes matrices), generalizing and simplifying an argument given in our previous work.
Paper Structure (17 sections, 29 theorems, 102 equations)

This paper contains 17 sections, 29 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Topological spaces and manifolds
  4. Holonomic D-modules
  5. Sheaves
  6. Ind-sheaves and subanalytic sheaves
  7. Enhanced ind-sheaves
  8. Riemann--Hilbert correspondences for analytic D-modules
  9. Galois conjugation
  10. Conjugation of D-modules and the enhanced de Rham functor
  11. The Hermitian dual
  12. Hermitian duality and the enhanced de Rham complex
  13. Galois descent for enhanced ind-sheaves
  14. $K$-structures and monodromy data
  15. Descent of meromorphic normal form
  16. ...and 2 more sections

Key Result

Lemma 2.2

Let $\mathrm{P}_X(\mathcal{M})$ be a statement concerning a complex manifold $X$ and an object $\mathcal{M}\in\mathrm{D}^\mathrm{b}_\mathrm{hol}(\mathcal{D}_{X})$. Then, $\mathrm{P}_X(\mathcal{M})$ is true for any complex manifold $X$ and any $\mathcal{M}\in \mathrm{D}^\mathrm{b}_\mathrm{hol}(\mathc

Theorems & Definitions (64)

  • Definition 2.1: cf. DK16
  • Lemma 2.2: cf. DK16
  • Lemma 2.3
  • Remark 2.4
  • Definition 2.5: cf. e.g. DKmicrolocal
  • Lemma 2.6
  • Remark 2.7
  • Lemma 3.1: see MocMT
  • Lemma 3.2
  • proof
  • ...and 54 more