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Generalized nearby cycles via relative and logarithmic $\mathscr{D}$-modules

Lei Wu

TL;DR

The work generalizes nearby-cycle theory for holonomic $\mathscr{D}$-modules to a logarithmic, relative setting governed by a graph-induced log structure on $X\times\mathbb{A}^r$, establishing a robust framework that links Bernstein–Sato ideals along monoid bases with Sabbah specialization complexes via a relative Riemann–Hilbert correspondence. Central contributions include the construction of generalized nearby-cycle modules, a logarithmic Artin–Rees theory, and two main comparison theorems: (i) a geometric description of the zero loci of Bernstein–Sato ideals as finite unions of translated linear subvarieties, and (ii) a precise equivalence between the analytic Sabbah specialization complex and the relative de Rham realization of generalized nearby cycles. The results extend Kashiwara–Malgrange-type filtrations to a relative logarithmic context, and they provide a geometric interpretation of $B$-ideals, with implications for Budur–Shi–Zuo-type questions and potential Budur-type conjectures in higher dimensions. The framework is applicable in particular to $\mathcal M=\mathcal O_X$, yielding a topological interpretation of Bernstein–Sato loci via exponential mappings and underlying $\mathbb{Q}$-mixed Hodge structures.

Abstract

For a regular map $F$ from a complex smooth affine variety $X$ to $\mathbb A^r_\mathbb C$, we construct generalized nearby-cycle modules of a regular holonomic $\mathscr D$-modules $\mathcal M$ along log strata with the log structure induced by the graph of $F$, whose relative supports are infinite unions of translated linear subvarieties of $\mathbb C^r$ determined by the zero loci of Bernstein-Sato ideals along monoid ideals. For a fixed log stratum, the nearby-cycle module corresponds to the Sabbah specialization complex of DR$(\mathcal M)$ under the relative regular Riemann-Hilbert correspondence of Fiorot-Fernandes-Sabbah, which generalizes the classical comparison theorem of Kashiwara-Malgrange for Deligne's nearby cycles. As an application, when $\mathcal M=\mathcal O_X$, we give a topological interpretation of the zero loci of Bernstein-Sato ideals of $F$ along monoid ideals under the exponential map, which answers a question of Budur-Shi-Zuo.

Generalized nearby cycles via relative and logarithmic $\mathscr{D}$-modules

TL;DR

The work generalizes nearby-cycle theory for holonomic -modules to a logarithmic, relative setting governed by a graph-induced log structure on , establishing a robust framework that links Bernstein–Sato ideals along monoid bases with Sabbah specialization complexes via a relative Riemann–Hilbert correspondence. Central contributions include the construction of generalized nearby-cycle modules, a logarithmic Artin–Rees theory, and two main comparison theorems: (i) a geometric description of the zero loci of Bernstein–Sato ideals as finite unions of translated linear subvarieties, and (ii) a precise equivalence between the analytic Sabbah specialization complex and the relative de Rham realization of generalized nearby cycles. The results extend Kashiwara–Malgrange-type filtrations to a relative logarithmic context, and they provide a geometric interpretation of -ideals, with implications for Budur–Shi–Zuo-type questions and potential Budur-type conjectures in higher dimensions. The framework is applicable in particular to , yielding a topological interpretation of Bernstein–Sato loci via exponential mappings and underlying -mixed Hodge structures.

Abstract

For a regular map from a complex smooth affine variety to , we construct generalized nearby-cycle modules of a regular holonomic -modules along log strata with the log structure induced by the graph of , whose relative supports are infinite unions of translated linear subvarieties of determined by the zero loci of Bernstein-Sato ideals along monoid ideals. For a fixed log stratum, the nearby-cycle module corresponds to the Sabbah specialization complex of DR under the relative regular Riemann-Hilbert correspondence of Fiorot-Fernandes-Sabbah, which generalizes the classical comparison theorem of Kashiwara-Malgrange for Deligne's nearby cycles. As an application, when , we give a topological interpretation of the zero loci of Bernstein-Sato ideals of along monoid ideals under the exponential map, which answers a question of Budur-Shi-Zuo.
Paper Structure (21 sections, 54 theorems, 338 equations)

This paper contains 21 sections, 54 theorems, 338 equations.

Table of Contents

  1. Introduction
  2. Log structures and Bernstein-Sato ideals
  3. Generalized nearby-cycle modules
  4. Geometric Budur-type conjecture
  5. Logarithmic $\mathscr{D}$-modules in the normal crossing case
  6. Basic set-up
  7. Pure filtrations of Gabber-Kashiwara
  8. Maximal tame pure extensions of lattices
  9. Analytic and algebraic relative regular holonomic $\mathscr{D}$-modules
  10. Relative $\mathbb{C}$-constructible sheaves
  11. Analytic relative regular holonomic $\mathscr{D}$-modules
  12. Essential fibers of relative holonomic $\mathscr{D}$-modules
  13. $\mathbb{Q}$-specializable $\mathscr{D}$-modules
  14. Relative minimal extensions and generalized nearby cycles
  15. Geometry of relative supports
  16. ...and 6 more sections

Key Result

Theorem A

With notation as above, let $\mathcal{M}$ be a regular holonomic $\mathscr{D}_X$-module. Then, for $K\subseteq \mathbb{N}^r$ and $\bm\in \mathbb{N}^r$, is a finite union of translated subtori, where Exp$:\mathbb{C}^r\to (\mathbb{C}^*)^r$ is the universal covering map. Furthermore, if $\mathcal{M}$ underlines a $\mathbb{Q}$-mixed Hodge module, then is a finite union of torsion translated subtori.

Theorems & Definitions (108)

  • Theorem A
  • Corollary B
  • Theorem C
  • Theorem D
  • Conjecture E: Geometric Budur-type Conjecture
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 98 more