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Fourier transforms of irregular holonomic D-modules, singularities at infinity of meromorphic functions and irregular characteristic cycles

Kiyoshi Takeuchi

TL;DR

This work advances the theory of Fourier transforms for holonomic D-modules by treating irregular, higher-dimensional cases. It combines irregular Riemann–Hilbert correspondence with enhanced ind-sheaves and meromorphic vanishing cycles to geometrize exponential data, including irregular micro-support and irregular characteristic cycles. A central result is that, for modules twisted by an exponential factor $f= rac{P}{Q}$, there exists a nonempty open set where the Fourier transform is a rank-$r$ algebraic connection, with $r$ determined by multiplicities of meromorphic vanishing cycles; the irregular data are controlled by a stationary-phase-type analysis and the irregular characteristic cycle CC$_{ ext{irr}}( rak M)$. Rank-jump phenomena can occur due to singularities of linear perturbations $f^w$ at indeterminacy points, reflecting richer higher-dimensional behavior than in the dimension-one case. Overall, the paper extends irregular Fourier analysis from dimension one to higher dimensions and provides a geometric framework for exponential factors, irregularities, and their microlocal invariants.

Abstract

Based on the recent developments in the irregular Riemann-Hilbert correspondence for holonomic D-modules and the Fourier-Sato transforms for enhanced ind-sheaves, we study the Fourier transforms of some irregular holonomic D-modules. For this purpose, the singularities of rational and meromorphic functions on complex affine varieties will be studied precisely, with the help of some new methods and tools such as meromorphic vanishing cycle functors. As a consequence, we show that the exponential factors and the irregularities of the Fourier transform of a holonomic D-module are described geometrically by the stationary phase method, as in the classical case of dimension one. A new feature in the higher-dimensional case is that we have some extra rank jump of the Fourier transform produced by the singularities of the linear perturbations of the exponential factors at their points of indeterminacy. In the course of our study, not necessarily homogeneous Lagrangian cycles that we call irregular characteristic cycles will play a crucial role.

Fourier transforms of irregular holonomic D-modules, singularities at infinity of meromorphic functions and irregular characteristic cycles

TL;DR

This work advances the theory of Fourier transforms for holonomic D-modules by treating irregular, higher-dimensional cases. It combines irregular Riemann–Hilbert correspondence with enhanced ind-sheaves and meromorphic vanishing cycles to geometrize exponential data, including irregular micro-support and irregular characteristic cycles. A central result is that, for modules twisted by an exponential factor , there exists a nonempty open set where the Fourier transform is a rank- algebraic connection, with determined by multiplicities of meromorphic vanishing cycles; the irregular data are controlled by a stationary-phase-type analysis and the irregular characteristic cycle CC. Rank-jump phenomena can occur due to singularities of linear perturbations at indeterminacy points, reflecting richer higher-dimensional behavior than in the dimension-one case. Overall, the paper extends irregular Fourier analysis from dimension one to higher dimensions and provides a geometric framework for exponential factors, irregularities, and their microlocal invariants.

Abstract

Based on the recent developments in the irregular Riemann-Hilbert correspondence for holonomic D-modules and the Fourier-Sato transforms for enhanced ind-sheaves, we study the Fourier transforms of some irregular holonomic D-modules. For this purpose, the singularities of rational and meromorphic functions on complex affine varieties will be studied precisely, with the help of some new methods and tools such as meromorphic vanishing cycle functors. As a consequence, we show that the exponential factors and the irregularities of the Fourier transform of a holonomic D-module are described geometrically by the stationary phase method, as in the classical case of dimension one. A new feature in the higher-dimensional case is that we have some extra rank jump of the Fourier transform produced by the singularities of the linear perturbations of the exponential factors at their points of indeterminacy. In the course of our study, not necessarily homogeneous Lagrangian cycles that we call irregular characteristic cycles will play a crucial role.
Paper Structure (15 sections, 72 theorems, 568 equations, 1 table)

This paper contains 15 sections, 72 theorems, 568 equations, 1 table.

Table of Contents

  1. Introduction
  2. Meromorphic Nearby and Vanishing Cycle Functors
  3. Singularities at Infinity of Meromorphic Functions
  4. An Overview on Ind-sheaves and the Irregular Riemann-Hilbert Correspondence
  5. Ind-sheaves
  6. Ind-sheaves on Bordered Spaces
  7. Enhanced Sheaves
  8. Enhanced Ind-sheaves
  9. D-modules
  10. Several Micro-supports Related to Holonomic D-modules
  11. Fourier Transforms of Exponentially Twisted Holonomic D-modules
  12. Toward the Study of Fourier Transforms of General Holonomic D-modules
  13. Preliminary Results for Holonomic D-modules
  14. Fourier Transforms of Standard Holonomic D-modules
  15. Some Auxiliary Results

Key Result

Theorem 1.1

Let $\mathcal{M}$ be an algebraic regular holonomic D-module on $X= \mathbb{C}^N$. Assume that its solution complex $Sol_{X}(\mathcal{M})$ is monodromic. Then its Fourier transform $\mathcal{M}^\wedge$ is regular and $Sol_{Y}(\mathcal{M}^\wedge)$ is monodromic.

Theorems & Definitions (120)

  • Theorem 1.1: (Brylinski Bry86)
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.4
  • proof
  • Lemma 2.5
  • ...and 110 more