Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The microlocal Riemann-Hilbert correspondence for complex contact manifolds

Laurent Côté, Christopher Kuo, David Nadler, Vivek Shende

TL;DR

The paper proves a global microlocal Riemann–Hilbert correspondence for complex contact manifolds, establishing an equivalence between perverse microsheaves and regular holonomic modules over Kashiwara's microlocal differential operators $ ext{E}_V$. The core construction uses Maslov data to glue local microsheaf theories across complex Darboux charts, leveraging large-codimension embeddings and GKS quantization to produce canonical kernels realizing chart-to-chart equivalences, and then verifying compatibility with Verdier duality. A parallel microlocal RH picture is developed for twisted local systems and twisted $ ext{D}$-modules, with a detailed Morita-theoretic and Picard-good framework ensuring that twists, dualities, and kernel constructions behave coherently under gluing. The results connect to deformation-quantization formalisms via $ ext{W}$-modules and $F$-actions, and illuminate links to representation-theoretic structures such as Fukaya-type categories through microlocal sheaf theory. Altogether, the work provides a robust, global microlocal dictionary between twisted microsheaves and microlocal $ ext{E}$-modules on complex contact manifolds, with a strong structural backbone given by Maslov data and Verdier duality.

Abstract

Kashiwara showed in 1996 that the categories of microlocalized D-modules can be canonically glued to give a sheaf of categories over a complex contact manifold. Much more recently, and by rather different considerations, we constructed a canonical notion of perverse microsheaves on the same class of spaces. Here we provide a Riemann-Hilbert correspondence.

The microlocal Riemann-Hilbert correspondence for complex contact manifolds

TL;DR

The paper proves a global microlocal Riemann–Hilbert correspondence for complex contact manifolds, establishing an equivalence between perverse microsheaves and regular holonomic modules over Kashiwara's microlocal differential operators . The core construction uses Maslov data to glue local microsheaf theories across complex Darboux charts, leveraging large-codimension embeddings and GKS quantization to produce canonical kernels realizing chart-to-chart equivalences, and then verifying compatibility with Verdier duality. A parallel microlocal RH picture is developed for twisted local systems and twisted -modules, with a detailed Morita-theoretic and Picard-good framework ensuring that twists, dualities, and kernel constructions behave coherently under gluing. The results connect to deformation-quantization formalisms via -modules and -actions, and illuminate links to representation-theoretic structures such as Fukaya-type categories through microlocal sheaf theory. Altogether, the work provides a robust, global microlocal dictionary between twisted microsheaves and microlocal -modules on complex contact manifolds, with a strong structural backbone given by Maslov data and Verdier duality.

Abstract

Kashiwara showed in 1996 that the categories of microlocalized D-modules can be canonically glued to give a sheaf of categories over a complex contact manifold. Much more recently, and by rather different considerations, we constructed a canonical notion of perverse microsheaves on the same class of spaces. Here we provide a Riemann-Hilbert correspondence.

Paper Structure

This paper contains 36 sections, 104 theorems, 272 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Twisted sheaves and microlocalization
  3. Linearity
  4. Twisting
  5. Twisted sheaves
  6. Microlocalization for twisted sheaves
  7. Microlocalizing integral transforms
  8. Integral transforms and Maslov data
  9. Maslov data
  10. Polarization-to-kernel
  11. Large codimensional embeddings
  12. GKS quantization
  13. Some remarks on principal bundles
  14. From polarization-to-kernel to Maslov-to-kernel
  15. Verdier duality on contact manifolds
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $V$ be a complex contact manifold. There is a canonical equivalence between perverse microsheaves on $V$ (in the sense of CKNS) and regular holonomic $\mathcal{E}_V$-modules (in the sense of kashiwara-quantization-contact) extending the microlocal Riemann--Hilbert correspondence.

Figures (1)

  • Figure 1: Čech 3-cocycle condition

Theorems & Definitions (247)

  • Theorem 1.1: \ref{['thm:gloabl mrh']}
  • Theorem 1.2: \ref{['thm: w-modules-to-mcsh']}
  • Remark 1.3
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 237 more