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On characteristic cycles of irregular holonomic D-modules

Kazuki Kudomi, Kiyoshi Takeuchi

Abstract

Based on the recent progress in the irregular Riemann-Hilbert correspondence for holonomic D-modules, we show that the characteristic cycles of some standard irregular holonomic D-modules can be expressed as in the classical theorem of Ginsburg. For this purpose, we first prove a formula for the enhanced solution complexes of holonomic D-modules having a quasi-normal form, via which, to our surprise, their solution complexes can be calculated more easily by topological methods. In the formulation and the proof of our main theorems, not necessarily homogeneous Lagrangian cycles that we call irregular characteristic cycles will play a crucial role.

On characteristic cycles of irregular holonomic D-modules

Abstract

Based on the recent progress in the irregular Riemann-Hilbert correspondence for holonomic D-modules, we show that the characteristic cycles of some standard irregular holonomic D-modules can be expressed as in the classical theorem of Ginsburg. For this purpose, we first prove a formula for the enhanced solution complexes of holonomic D-modules having a quasi-normal form, via which, to our surprise, their solution complexes can be calculated more easily by topological methods. In the formulation and the proof of our main theorems, not necessarily homogeneous Lagrangian cycles that we call irregular characteristic cycles will play a crucial role.

Paper Structure

This paper contains 14 sections, 25 theorems, 213 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary Notions and Results
  3. Ind-sheaves
  4. Ind-sheaves on Bordered Spaces
  5. Enhanced Sheaves
  6. Enhanced Ind-sheaves
  7. $\mathcal{D}$-Modules
  8. Constructible functions and constructible sheaves
  9. Solution complexes to irregular holonomic D-modules with a quasi-normal form
  10. An index formula for irregular connections
  11. Ginsburg type formulas for characteristic cycles
  12. A supplement to the proof of Proposition \ref{['prop-1']}
  13. The case of $d=1$
  14. The computation of $H^\ast((S^1)^l; \mathbb{C}_{W_{k_1,\dots,k_l}})$

Key Result

Theorem 1.1

In the situation as above, let $g\colon U\longrightarrow\mathbb{C}$ be a defining holomorphic function of the normal crossing divisor $D\cap U\subset U$. Then in the open subset $T^\ast U\subset T^\ast X$ we have where the limit in the right hand side stands for that of Lagrangian cycles (see FKT26).

Theorems & Definitions (44)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2: DK16
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Corollary 3.3
  • ...and 34 more