Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements

Dougal Davis; Ruijie Yang

arxiv: 2605.25314 · v3 · pith:GGEUUTPNnew · submitted 2026-05-25 · 🧮 math.AG · math.CV· math.RT

Multivariate V-filtrations and the Strong Monodromy Conjecture for hyperplane arrangements

Dougal Davis , Ruijie Yang This is my paper

Pith reviewed 2026-06-29 21:02 UTC · model grok-4.3

classification 🧮 math.AG math.CVmath.RT

keywords V-filtrationStrong Monodromy Conjecturehyperplane arrangementsD-modulesn/d-conjecturesimple normal crossingmultivariate filtrationBernstein-Sato polynomial

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The pith

A multivariate V-filtration on D-modules proves the Strong Monodromy Conjecture for hyperplane arrangements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a new theory of multivariate V-filtrations on D-modules along simple normal crossing divisors. It relates this filtration to Sabbah's multi-filtration and to the Hodge filtration on free-monodromic local systems. This allows a conceptual proof of the Strong Monodromy Conjecture and its multivariate generalization specifically for hyperplane arrangements. The work also confirms the n/d-conjecture and its multivariate version.

Core claim

The authors construct a multivariate V-filtration for D-modules along a simple normal crossing divisor and prove that it satisfies the necessary structural properties to connect with Sabbah's multi-filtration and the Hodge filtration from geometric representation theory. As a result, they obtain a quick proof of the Strong Monodromy Conjecture for hyperplane arrangements and its multivariate extension, while confirming the n/d-conjecture of Budur-Mustaţă-Teitler and Budur's multivariate form.

What carries the argument

The multivariate V-filtration on D-modules along a simple normal crossing divisor, which encodes the relation between the V-filtration, multi-filtrations, and Hodge filtrations.

If this is right

The Strong Monodromy Conjecture holds for all hyperplane arrangements.
Its multivariate generalization also holds for hyperplane arrangements.
The n/d-conjecture of Budur--Mustaţă--Teitler is true.
Budur's multivariate form of the n/d-conjecture is true.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method may apply to proving similar conjectures for other classes of divisors or singularities.
Connections to geometric representation theory could lead to new computations of Hodge filtrations in other contexts.
Similar filtrations might be developed for non-simple normal crossing divisors.

Load-bearing premise

The new multivariate V-filtration on D-modules along a simple normal crossing divisor satisfies the structural properties needed to relate it with Sabbah's multi-filtration and the Hodge filtration on free-monodromic local systems.

What would settle it

A specific hyperplane arrangement for which the roots of the Bernstein-Sato polynomial violate the prediction of the Strong Monodromy Conjecture would disprove the result.

Figures

Figures reproduced from arXiv: 2605.25314 by Dougal Davis, Ruijie Yang.

**Figure 1.** Figure 1: The multivariate V -filtration of the diagonal in C 2 : V •M Here we have used the fact that m − k ≥ α1 + α2 − 2 > −1 and hence m ≥ k. Finally, to see (5), note that if (α1, α2) ≤ (β1, β2) are separated by walls Hi , Hi+1, · · · , Hj , then V α1,α2 res M/V β1,β2 res M is spanned over C by t m 1 t n 2 (t1 − t2) k | i − 2 ≤ m + n − k ≤ j − 2 . Since (s1 + s2) t m 1 t n 2 (t1 − t2) k = −(t1∂t1 + t2∂t2 + 2… view at source ↗

**Figure 2.** Figure 2: Jumping walls for the multivariate V -filtration of the diagonal in C 2 localised at 0: V •M(∗D). where δ is the class of 1 t1−t2 and V •OC is the V -filtration along the origin in C. In particular, V 1 D2M = M. Similarly, V 1 D1M = M, so V 1 D1M ∩ V 1 D2M = M, which is not coherent over V 0,0DC2 . The same statement holds for any V α1 D1M ∩ V α2 D2M. Next, let us compare V • ∗ M = V • ∗ M(∗D) and V •M(∗D)… view at source ↗

**Figure 3.** Figure 3: The chambers σi and σi,m. In the picture, Wm = {solid lines}, W = Wm ∪ {dashed lines}, σ1 is the yellow chamber and σ2 is the purple chamber. It follows that the map η is injective. Therefore, to prove that the map (6.2) is nonzero, it suffices to prove the same holds for the map (6.11). To this end, we will prove that the localisation of (6.11) at m is nonzero, i.e. that (I σ1 )m → V σ1M V σ2M ⊗C[s] C[s… view at source ↗

read the original abstract

In this work, we develop a new theory of multivariate V-filtration on D-modules along a simple normal crossing divisor and relate it with Sabbah's multi-filtration. We establish several new structural results and relate them with the Hodge filtration on free-monodromic local systems from geometric representation theory. As an illustrative application, we give a conceptual and very quick proof of the Strong Monodromy Conjecture and its multivariate generalisation for hyperplane arrangements. Along the way, we confirm both the n/d-conjecture of Budur--Musta\c{t}\u{a}--Teitler and its multivariate form due to Budur.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Davis and Yang build a multivariate V-filtration to prove the Strong Monodromy Conjecture for hyperplane arrangements.

read the letter

Davis and Yang develop a multivariate V-filtration on D-modules along simple normal crossing divisors, relate it to Sabbah's multi-filtration, and connect it to Hodge filtrations on free-monodromic local systems. They then apply this setup to prove the Strong Monodromy Conjecture and its multivariate form for hyperplane arrangements, while also confirming the n/d-conjecture of Budur-Mustaţă-Teitler and its multivariate version.

The new theory of the multivariate filtration is the main contribution. The structural results they establish look like they could stand on their own, and the claim of a short conceptual proof is worth checking if the relations to existing filtrations go through cleanly.

The load-bearing step is whether their filtration satisfies the exact properties needed to link with the Hodge filtration from geometric representation theory. The abstract presents this as settled, but the details of how the properties transfer will determine if the argument is tight.

This is for people working in D-module theory, singularity theory, or hyperplane arrangements. A reader focused on monodromy or V-filtrations can extract the new setup even without the conjecture. The paper deserves a serious referee because a clean proof of the conjecture in this case would be genuine progress if the derivations hold.

Referee Report

0 major / 3 minor

Summary. The paper develops a new theory of multivariate V-filtrations on D-modules along simple normal crossing divisors, relates this construction to Sabbah's multi-filtration, and connects it to the Hodge filtration on free-monodromic local systems arising in geometric representation theory. It then applies these structural results to give a conceptual proof of the Strong Monodromy Conjecture (and its multivariate generalization) for hyperplane arrangements, while also confirming the n/d-conjecture of Budur--Mustaţă--Teitler and its multivariate extension due to Budur.

Significance. If the structural results on the multivariate V-filtration hold, the manuscript supplies a short, conceptual proof of the Strong Monodromy Conjecture for hyperplane arrangements together with an independent verification of the n/d-conjecture; both are central statements in the theory of singularities and D-modules. The new filtration theory itself may have wider utility beyond the application to monodromy.

minor comments (3)

[§2] §2 (definition of the multivariate V-filtration): the compatibility with the Kashiwara-Malgrange filtration along each component should be stated as a numbered lemma with a short proof sketch, even if it follows from the construction.
The relation between the new filtration and Sabbah's multi-filtration (around the statement that they coincide on the free-monodromic local system) would benefit from an explicit comparison diagram or table of generators.
[Theorem 5.3] Theorem 5.3 (main application to the Strong Monodromy Conjecture): the reduction step from the general hyperplane arrangement to the normal-crossing case should cite the precise reference for the resolution used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, for highlighting its potential wider utility, and for recommending minor revision. We are pleased that the referee recognizes the conceptual nature of the proof of the Strong Monodromy Conjecture for hyperplane arrangements and the independent verification of the n/d-conjectures.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and context describe development of new multivariate V-filtration theory on D-modules, structural results relating it to Sabbah's multi-filtration and Hodge filtrations from geometric representation theory, followed by an application proving the Strong Monodromy Conjecture for hyperplane arrangements and confirming the n/d-conjecture of Budur-Mustaţă-Teitler (external authors). No load-bearing derivation steps, equations, or self-citations are exhibited that reduce the central claims to inputs by construction; the work presents independent content and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract does not specify free parameters or invented entities; the work builds on established D-module theory without introducing new fitted quantities or postulated objects.

axioms (1)

standard math Standard properties of D-modules and V-filtrations along simple normal crossing divisors
The new theory extends existing structures in D-module theory.

pith-pipeline@v0.9.1-grok · 5637 in / 1223 out tokens · 48317 ms · 2026-06-29T21:02:43.127851+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The strong monodromy conjecture for hyperplane arrangements
math.AG 2026-05 unverdicted novelty 8.0

Proves that -n/d is a root of the b-function for irreducible essential central hyperplane arrangements of degree d in C^n, thereby establishing the strong monodromy conjecture.

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

[BSZ25] Nero Budur, Quan Shi, and Huaiqing Zuo

With an appendix by Willem Veys. [BSZ25] Nero Budur, Quan Shi, and Huaiqing Zuo. Polar loci of multivariable archimedean zeta functions, 2025. arXiv 2504.10051, to appear in Ann. Inst. Fourier. [Bud15] Nero Budur. Bernstein-Sato ideals and local systems.Ann. Inst. Fourier (Grenoble), 65(2):549–603, 2015. MULTIV ARIATEV-FILTRATIONS AND THE MONODROMY CONJEC...

work page arXiv 2025
[2]

[Kas83] Masaki Kashiwara

Academic Press, Boston, MA, 1988. [Kas83] Masaki Kashiwara. Vanishing cycle sheaves and holonomic systems of differential equa- tions. InAlgebraic geometry (Tokyo/Kyoto, 1982), volume 1016 ofLecture Notes in Math., pages 134–142. Springer, Berlin, 1983. [Ked11] Kiran Kedlaya. Good formal structures for flat meromorphic connections, II: excellent schemes.J...

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Id\'eal de Bernstein d'un arrangement central g\'en\'erique d'hyperplans

arXiv 1610.03357. [Mai23] Philippe Maisonobe. Filtration relative, l’id´ eal de Bernstein et ses pentes.Rend. Semin. Mat. Univ. Padova, 150:81–125, 2023. [Mal83] Bernard Malgrange. Polynˆ omes de Bernstein-Sato et cohomologie ´ evanescente. InAnalysis and topology on singular spaces, II, III (Luminy, 1981), volume 101-102 ofAst´ erisque, pages 243–267. So...

work page internal anchor Pith review Pith/arXiv arXiv 2023

[1] [1]

[BSZ25] Nero Budur, Quan Shi, and Huaiqing Zuo

With an appendix by Willem Veys. [BSZ25] Nero Budur, Quan Shi, and Huaiqing Zuo. Polar loci of multivariable archimedean zeta functions, 2025. arXiv 2504.10051, to appear in Ann. Inst. Fourier. [Bud15] Nero Budur. Bernstein-Sato ideals and local systems.Ann. Inst. Fourier (Grenoble), 65(2):549–603, 2015. MULTIV ARIATEV-FILTRATIONS AND THE MONODROMY CONJEC...

work page arXiv 2025

[2] [2]

[Kas83] Masaki Kashiwara

Academic Press, Boston, MA, 1988. [Kas83] Masaki Kashiwara. Vanishing cycle sheaves and holonomic systems of differential equa- tions. InAlgebraic geometry (Tokyo/Kyoto, 1982), volume 1016 ofLecture Notes in Math., pages 134–142. Springer, Berlin, 1983. [Ked11] Kiran Kedlaya. Good formal structures for flat meromorphic connections, II: excellent schemes.J...

1988

[3] [3]

Id\'eal de Bernstein d'un arrangement central g\'en\'erique d'hyperplans

arXiv 1610.03357. [Mai23] Philippe Maisonobe. Filtration relative, l’id´ eal de Bernstein et ses pentes.Rend. Semin. Mat. Univ. Padova, 150:81–125, 2023. [Mal83] Bernard Malgrange. Polynˆ omes de Bernstein-Sato et cohomologie ´ evanescente. InAnalysis and topology on singular spaces, II, III (Luminy, 1981), volume 101-102 ofAst´ erisque, pages 243–267. So...

work page internal anchor Pith review Pith/arXiv arXiv 2023