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Riemann-Hilbert correspondence for Alexander complexes

Lei Wu

TL;DR

The work provides a global, functorial Riemann–Hilbert correspondence for Alexander (Sabbah) complexes in a relative setting by leveraging $G$-equivariant relative holonomic $ ext{D}$-modules and maximal/minimal extensions along a holomorphic map $F=(f_1, o,f_r)$. It connects the relative de Rham complex of $oldsymbol{ ext{Ψ}}_F( ext{M})$ with Sabbah’s specialization complex via a pushforward along the universal cover, and describes the relative supports and characteristic cycles through Bernstein–Sato ideals and monodromy data. The paper also develops a comprehensive framework for relative sheafification, duality, and codimension filtrations, enabling precise statements about the geometry of supports and the behavior under base change. Together, these results generalize known local pictures to a global, equivariant, relative Riemann–Hilbert perspective with explicit formulas for monodromy and a linearity conjecture guiding future work on the structure of relative supports.

Abstract

We establish a relative Riemann-Hilbert correspondence for Alexander complexes (also known as Sabbah specialization complexes) by using relative regular holonomic $\mathscr D$-modules in an equivariant way, which particularly gives a "global" approach to the correspondence for Deligne's nearby cycles. Using the correspondence and zero loci of Bernstein-Sato ideals, we obtain a formula for the relative support of the Alexander complexes.

Riemann-Hilbert correspondence for Alexander complexes

TL;DR

The work provides a global, functorial Riemann–Hilbert correspondence for Alexander (Sabbah) complexes in a relative setting by leveraging -equivariant relative holonomic -modules and maximal/minimal extensions along a holomorphic map . It connects the relative de Rham complex of with Sabbah’s specialization complex via a pushforward along the universal cover, and describes the relative supports and characteristic cycles through Bernstein–Sato ideals and monodromy data. The paper also develops a comprehensive framework for relative sheafification, duality, and codimension filtrations, enabling precise statements about the geometry of supports and the behavior under base change. Together, these results generalize known local pictures to a global, equivariant, relative Riemann–Hilbert perspective with explicit formulas for monodromy and a linearity conjecture guiding future work on the structure of relative supports.

Abstract

We establish a relative Riemann-Hilbert correspondence for Alexander complexes (also known as Sabbah specialization complexes) by using relative regular holonomic -modules in an equivariant way, which particularly gives a "global" approach to the correspondence for Deligne's nearby cycles. Using the correspondence and zero loci of Bernstein-Sato ideals, we obtain a formula for the relative support of the Alexander complexes.

Paper Structure

This paper contains 30 sections, 41 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Riemann-Hilbert correspondence for Alexander complex
  3. Relative Support
  4. Relative characteristic cycle and monodromy zeta function
  5. Outline of the paper
  6. Relative maximal and minimal extensions along hypersurfaces
  7. Notations
  8. $G$-action and translation on $\mathscr{M}$
  9. Relative maximal and minimal extension
  10. Gluing data and $G$-equivariance
  11. $G$-equivariance of relative de Rham complex
  12. Bernstein-Sato ideal along $F$
  13. Evaluating $\Psi_F(\mathcal{M})$
  14. A linearity conjecture
  15. Alexander complex of Sabbah
  16. ...and 15 more sections

Key Result

Theorem 1.1

With the notation as above, if $\mathcal{M}$ is a regular holonomic $\mathscr{D}_X$-module, then we have a natural quasi-isomorphism where $\pi_*^G$ is the equivariant direct image functor (see BL), $\textup{DR}_{X\times\mathbb{C}^r/\mathbb{C}^r}$ is the relative de Rham functor on $X\times\mathbb{C}^r$ over $\mathbb{C}^r$ and $\widetilde{\psi}_{F}(\textup{DR}_X(\mathcal{M}))$ is the analytic she

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 2.1: Sabbah
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • ...and 70 more