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Enhanced nearby and vanishing cycles in dimension one and Fourier transform

Andrea D'Agnolo, Masaki Kashiwara

Abstract

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, and denote by ${}^{\mathsf{L}}\mathcal M$ its Fourier-Laplace transform. For a point $a$ on the affine line, denote by $\ell_a$ the corresponding linear function on the dual affine line. Then, the vanishing cycles of $\mathcal M$ at $a$ are isomorphic to the graded component of degree $\ell_a$ of the Stokes filtration of ${}^{\mathsf{L}}\mathcal M$ at infinity.

Enhanced nearby and vanishing cycles in dimension one and Fourier transform

Abstract

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann-Hilbert correspondence. In this paper, we give some precisions on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let be a holonomic algebraic -module on the affine line, and denote by its Fourier-Laplace transform. For a point on the affine line, denote by the corresponding linear function on the dual affine line. Then, the vanishing cycles of at are isomorphic to the graded component of degree of the Stokes filtration of at infinity.

Paper Structure

This paper contains 19 sections, 16 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review on enhanced ind-sheaves
  3. Ind-sheaves and bordered spaces
  4. Enhanced ind-sheaves
  5. Specialization and microlocalization
  6. Constructibility
  7. Enhanced perverse ind-sheaves on a curve
  8. Normal form
  9. Perversity
  10. Nearby and vanishing cycles
  11. Definitions
  12. The case of perverse objects
  13. Fourier transform on the affine line
  14. Linear exponential factors
  15. Smash functor
  16. ...and 4 more sections

Key Result

Proposition 3.2

Let $K\in\mathrm{E}^{\mathrm{b}}_{\mathbb{R}\text{-}\mathrm{c}}(\mathrm{I}\mspace{2mu}\mathbf{k}_{X})$ have normal form at $I\subset S_a X$. Then $\mathrm{E}\nu^\mathsf{rb}_{{\{{a}\}}}(K)|_I$ is of sheaf type. More precisely, $\mathrm{E}\nu^\mathsf{rb}_{{\{{a}\}}}(K)|_I\simeq e\iota(L)$ for $L\in\ma

Figures (1)

  • Figure 1: The sets $\overline{j(U)}$, $\overline{\widetilde{\jmath}(U)}$ and $\overline{\widehat{\jmath}(U)}$ pictured in the case $M=\mathbb{R}$ and $N={\{{0}\}}$. The red lines are fibers of the projection $p_U\colon U\xrightarrow{}\Omega\xrightarrow{}M$. (Color figure online.)

Theorems & Definitions (36)

  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Theorem 3.6
  • Proposition 3.7
  • ...and 26 more