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Enhanced Perverse Subanalytic Sheaves

Yohei Ito

TL;DR

The paper constructs a t-structure on the triangulated category of $\mathbb{C}$-constructible enhanced subanalytic (and algebraic) sheaves, whose heart is equivalent to the abelian category of holonomic $\mathcal{D}$-modules. It unifies analytic and algebraic settings via bordered spaces and enhanced subanalytic frameworks, showing that holonomic $\mathcal{D}$-modules arise as the heart and that $\mathbb{C}$-constructible enhanced subanalytic objects correspond to perverse-type images of solutions (e.g., $${\rm Sol}_X^{\rm E,sub}$$) with explicit t-structure compatibilities. The results extend the irregular Riemann–Hilbert correspondence to new categorical contexts and provide tools to study simple objects and minimal extensions within enhanced perverse subanalytic sheaves. This advances the understanding of irregular singularities through subanalytic and bordered-space formalisms, enabling precise abelian and derived-category correspondences with holonomic $\mathcal{D}$-modules and their perverse shadows.

Abstract

In [arXiv:2109.13991], the author explained a relation between enhanced ind-sheaves and enhanced subanalytic sheaves. In particular, a relation between [Thm.9.5.3, Andrea D'Agnolo and Masaki Kashiwara, Riemann-Hilbert correspondence for holonomic $\mathcal{D}$-modules, 2016] and [Thm.6.3, Masaki Kashiwara, Riemann-Hilbert correspondence for irregular holonomic $\mathcal{D}$-modules, 2016] had been explained. Moreover, in [arXiv:2310.19501], the author defined $\mathbb{C}$-constructibility for enhanced subanalytic sheaves and proved that there exists an equivalence of categories between the triangulated category of holonomic $\mathcal{D}$-modules and that of $\mathbb{C}$-constructible enhanced subanalytic sheaves. In this paper, we will show that there exists a t-structure on the triangulated category of $\mathbb{C}$-constructible enhanced subanalytic sheaves whose heart is equivalent to the abelian category of holonomic $\mathcal{D}$-modules. Furthermore, we shall consider simple objects of its heart and minimal extensions of objects of its heart.

Enhanced Perverse Subanalytic Sheaves

TL;DR

The paper constructs a t-structure on the triangulated category of -constructible enhanced subanalytic (and algebraic) sheaves, whose heart is equivalent to the abelian category of holonomic -modules. It unifies analytic and algebraic settings via bordered spaces and enhanced subanalytic frameworks, showing that holonomic -modules arise as the heart and that -constructible enhanced subanalytic objects correspond to perverse-type images of solutions (e.g., ) with explicit t-structure compatibilities. The results extend the irregular Riemann–Hilbert correspondence to new categorical contexts and provide tools to study simple objects and minimal extensions within enhanced perverse subanalytic sheaves. This advances the understanding of irregular singularities through subanalytic and bordered-space formalisms, enabling precise abelian and derived-category correspondences with holonomic -modules and their perverse shadows.

Abstract

In [arXiv:2109.13991], the author explained a relation between enhanced ind-sheaves and enhanced subanalytic sheaves. In particular, a relation between [Thm.9.5.3, Andrea D'Agnolo and Masaki Kashiwara, Riemann-Hilbert correspondence for holonomic -modules, 2016] and [Thm.6.3, Masaki Kashiwara, Riemann-Hilbert correspondence for irregular holonomic -modules, 2016] had been explained. Moreover, in [arXiv:2310.19501], the author defined -constructibility for enhanced subanalytic sheaves and proved that there exists an equivalence of categories between the triangulated category of holonomic -modules and that of -constructible enhanced subanalytic sheaves. In this paper, we will show that there exists a t-structure on the triangulated category of -constructible enhanced subanalytic sheaves whose heart is equivalent to the abelian category of holonomic -modules. Furthermore, we shall consider simple objects of its heart and minimal extensions of objects of its heart.

Paper Structure

This paper contains 13 sections, 41 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Riemann--Hilbert Problem
  3. Regular Riemann--Hilbert Correspondence
  4. Perverse Sheaves
  5. Irregular Riemann--Hilbert Correspondence and Enhanced Ind-Sheaves
  6. $\mathbb{C}$-Constructible Enhanced Ind-Sheaves
  7. Algebraic Irregular Riemann--Hilbert Correspondence and Enhanced Ind-Sheaves
  8. Irregular Riemann--Hilbert Correspondence and Enhanced Subanalytic Sheaves
  9. Algebraic Irregular Riemann--Hilbert Correspondence and Enhanced Subanalytic Sheaves
  10. Main Result
  11. Analytic Case
  12. Algebraic Case
  13. Minimal Extensions

Key Result

Theorem 1.1

There exists an equivalence of categories:

Theorems & Definitions (73)

  • Theorem 1.1: Kas84
  • Theorem 1.2: DK16
  • Theorem 1.3: Ito20, Ito23
  • Theorem 1.4: Ito20
  • Theorem 1.5: Ito21
  • Theorem 1.6: Ito21
  • Theorem 1.7: Kas16
  • Theorem 1.8: Ito24a
  • Theorem 1.9: Ito24b
  • Theorem 1.10: Ito24b
  • ...and 63 more