Strong Monodromy Conjecture Overview

• The strong monodromy conjecture is a framework that asserts every pole of a zeta function is a root of the Bernstein–Sato polynomial, linking analytic and topological invariants.
• Variants of the conjecture, including motivic, multivariate, and meromorphic forms, generalize the relationship between zeta poles and monodromy eigenvalues in diverse settings.
• Techniques such as resolution of singularities, D-module theory, and combinatorial analyses are employed to verify the conjecture in cases like hyperplane arrangements, plane curves, and abelian varieties.

The Strong Monodromy Conjecture is a family of statements relating the poles of zeta functions attached to singularities or degenerations to monodromy, most commonly through the Bernstein–Sato polynomial. In the standard hypersurface formulation, it asserts that every pole $s_0$ of the local topological zeta function is a root of $b_f(s)$, equivalently that $\exp(2\pi i\,s_0)$ occurs as a local monodromy eigenvalue. Closely related motivic and multivariate versions replace the topological zeta function by Denef–Loeser’s motivic zeta function or by multivariable analogues, while for tamely ramified abelian varieties the “strong” form is sharper: the pole order is identified with the size of the maximal Jordan block for the corresponding monodromy eigenvalue (Davis et al., 25 May 2026, Bath et al., 24 Feb 2026, Halle et al., 2010).

1. Core formulations

For a polynomial $f$ on $X=\mathbb{C}^n$, the local topological zeta function is computed from an embedded resolution of $\{f=0\}$ by

$$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$

where $E_i$ are exceptional divisors of a log-resolution, $N_i\in\mathbb{Z}_{>0}$ is the vanishing order of $f\circ\pi$ along $b_f(s)$0, $b_f(s)$1 is the discrepancy, and $b_f(s)$2. The Bernstein–Sato polynomial $b_f(s)$3 is the monic polynomial of least degree satisfying

$b_f(s)$4

Malgrange–Kashiwara identify the roots of $b_f(s)$5 with the rational numbers $b_f(s)$6 such that $b_f(s)$7 is an eigenvalue of local monodromy on the Milnor fiber. In this setting, the Strong Monodromy Conjecture states that if $b_f(s)$8 is a pole of $b_f(s)$9, then $\exp(2\pi i\,s_0)$0 is a root of $\exp(2\pi i\,s_0)$1; equivalently, $\exp(2\pi i\,s_0)$2 is a monodromy eigenvalue (Davis et al., 25 May 2026).

Denef–Loeser’s motivic zeta function replaces $\exp(2\pi i\,s_0)$3 by a formal series

$\exp(2\pi i\,s_0)$4

with a resolution-theoretic expression

$\exp(2\pi i\,s_0)$5

Its motivic strong form is

$\exp(2\pi i\,s_0)$6

There is also a local version at the origin and a “stronger” statement predicting that the order of a pole is bounded by its multiplicity as a root (Bath et al., 24 Feb 2026).

A multivariate extension is formulated for a tuple $\exp(2\pi i\,s_0)$7 using the multivariate topological zeta function

$\exp(2\pi i\,s_0)$8

and the Bernstein–Sato ideal

$\exp(2\pi i\,s_0)$9

The multivariate conjecture predicts that every pole lies in the zero-set of $f$0 (Davis et al., 25 May 2026).

The broader monodromy package also includes $f$1-adic analogues, where one studies

$f$2

and conjectures the same implication “pole $f$3 root of $f$4” for almost all primes $f$5 (Davis et al., 25 May 2026).

2. Plane germs and the meromorphic extension

For plane meromorphic germs, the relevant object is a germ

$f$6

on $f$7, with $f$8 holomorphic and without common factor. An embedded resolution $f$9 is required to resolve the total transform of $X=\mathbb{C}^n$0 into a normal-crossing divisor and to extend $X=\mathbb{C}^n$1 holomorphically to $X=\mathbb{C}^n$2. For $X=\mathbb{C}^n$3, the associated $X=\mathbb{C}^n$4-Milnor fiber $X=\mathbb{C}^n$5 carries a monodromy operator $X=\mathbb{C}^n$6, and the corresponding monodromy zeta function is

$X=\mathbb{C}^n$7

The local topological zeta function of $X=\mathbb{C}^n$8 at $X=\mathbb{C}^n$9 is defined by

$\{f=0\}$0

where $\{f=0\}$1, $\{f=0\}$2 is the multiplicity datum from $\{f=0\}$3, and the definition is independent of the chosen resolution (Villa et al., 2013).

In the meromorphic plane case, the pole structure differs from the holomorphic case. The candidate poles are $\{f=0\}$4 for components with $\{f=0\}$5, together with $\{f=0\}$6 from strict-transform components of $\{f=0\}$7. In dimension two, the holomorphic Veys–Loeser criterion says that $\{f=0\}$8 is an actual pole if and only if $\{f=0\}$9 meets at least three other components or is a strict-transform component. For meromorphic germs, the “only if” part remains valid, but the “if” direction fails in general (Villa et al., 2013).

The principal theorem states that if $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$0 is a germ of a meromorphic function on $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$1, then for every pole $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$2 of $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$3, the complex number $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$4 occurs as an eigenvalue of the local monodromy transformation $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$5 on the Milnor fiber $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$6 or at some nearby point of $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$7. The proof separates the strict-transform case from the exceptional-divisor case with $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$8, uses intersection-number relations generalizing Veys–Loeser combinatorics, invokes a corollary of Delgado–Maugendre on primitive subtrees in the resolution graph, and applies an Euler-characteristic count on trees of rational curves via Rodrigues’ Lemma. A specialization of A’Campo’s formula then produces the required monodromy eigenvalue (Villa et al., 2013).

This completes the verification of the generalized strong monodromy statement for plane meromorphic germs, complementing Loeser’s classical proof for plane holomorphic curves and extending the theory to $$Z_{f,\mathrm{top}}(s)=\sum_{I\subseteq\{1,\dots,m\}} \chi\bigl(E_I^\circ\bigr)\,\prod_{i\in I}\frac{1}{\nu_i+sN_i},$$9-type singularities (Villa et al., 2013).

3. Hyperplane arrangements

For a central hyperplane arrangement on $E_i$0,

$E_i$1

with total degree $E_i$2, one distinguishes the central, essential, and irreducible cases. Budur–Mustaţă–Teitler identified the candidate pole coming from the smallest stratum as $E_i$3, and conjectured that for an irreducible, essential, central arrangement of degree $E_i$4 in $E_i$5,

$E_i$6

They also proved that this statement implies the strong monodromy conjecture for all poles of $E_i$7 (Wu, 25 May 2026).

Wu proved a stronger theorem: for every integer $E_i$8 with $E_i$9,

$N_i\in\mathbb{Z}_{>0}$0

In particular $N_i\in\mathbb{Z}_{>0}$1, so the Budur–Mustaţă–Teitler conjecture holds, and hence the strong monodromy conjecture for hyperplane arrangements is established. The proof uses algebraic relative $N_i\in\mathbb{Z}_{>0}$2-modules, the commutation of duality with proper direct image, Beilinson–Bernstein nearby cycles, the wonderful model of De Concini–Procesi as a canonical log-resolution, and a non-vanishing argument identifying a de Rham complex with the extension by zero of a rank $N_i\in\mathbb{Z}_{>0}$3 local system on the projective arrangement complement, followed by the propagation theorem of Denham–Suciu–Yuzvinsky (Wu, 25 May 2026).

A complementary development treats both the univariate and multivariate forms for arrangements. If $N_i\in\mathbb{Z}_{>0}$4 defines a central arrangement and $N_i\in\mathbb{Z}_{>0}$5 is its intersection lattice, then Budur–Mustaţă–Teitler showed that

$N_i\in\mathbb{Z}_{>0}$6

Davis and Yang proved: (i) every pole of $N_i\in\mathbb{Z}_{>0}$7 is a root of $N_i\in\mathbb{Z}_{>0}$8; (ii) for any factorization $N_i\in\mathbb{Z}_{>0}$9 into linear factors, every pole of the multivariate zeta $f\circ\pi$0 lies in the zero-set of the Bernstein–Sato ideal $f\circ\pi$1. Their proof develops a multivariate $f\circ\pi$2-filtration along a simple normal crossing divisor, proves uniqueness, strictness, functoriality, equivalence with Sabbah’s construction, relative holonomicity and duality, and a “magic formula” identifying $f\circ\pi$3 with Hodge-filtered pro-objects of the form $f\circ\pi$4. In the arrangement case, potential poles become rational walls of the multivariate $f\circ\pi$5-filtration, and jumps in those walls force roots of the classical Bernstein–Sato polynomial or zeros of the Bernstein–Sato ideal (Davis et al., 25 May 2026).

4. Homogeneous polynomials and projective curves in three variables

For a reduced projective curve $f\circ\pi$6 defined by $f\circ\pi$7 and having only weighted-homogeneous singularities, the strong monodromy conjecture can be proved by combining a three-variable Denef–Loeser formula with two-variable results. In suitable local analytic coordinates $f\circ\pi$8 at a singular point $f\circ\pi$9, a weighted-homogeneous singularity has local equation

$b_f(s)$00

with all monomials satisfying

$b_f(s)$01

Such singularities are Newton nondegenerate, their embedded resolution graphs are star-shaped, and the local monodromy eigenvalues are roots of unity $b_f(s)$02 for certain integers $b_f(s)$03. In the global three-variable Denef–Loeser formula,

$b_f(s)$04

candidate poles that come from faces lying in coordinate hyperplanes coincide with poles of two-variable local topological zeta functions and are therefore already known to satisfy the strong conjecture. In the extremely degenerate case there is a unique compact edge $b_f(s)$05, and a direct computation shows that the only new candidate pole,

$b_f(s)$06

cancels: the numerator of the total sum is divisible by $b_f(s)$07, so the pole does not appear. The only surviving poles are exactly those coming from local singularities of $b_f(s)$08, and hence every pole is a root of the global Bernstein–Sato polynomial $b_f(s)$09 (Saito, 27 Feb 2026).

A parallel motivic result holds for homogeneous, possibly non-reduced, polynomials whose associated reduced projective divisor $b_f(s)$10 has only quasi-homogeneous isolated singularities. In arbitrary dimension, Bath and Veys characterize when $b_f(s)$11 is a root of $b_f(s)$12 using elementary data involving logarithmic derivations. Writing

$b_f(s)$13

they show, under the quasi-homogeneous isolated-singularity hypothesis, that

$b_f(s)$14

if and only if $b_f(s)$15 and $b_f(s)$16 contains a semi-simple non-traceless derivation. In dimension three, after a homogeneous linear change of variables, any reduced, semi-simple $b_f(s)$17-symmetric plane curve can be written in the semi-simple standard form

$b_f(s)$18

with the non-traceless condition

$b_f(s)$19

Theorem A states that for homogeneous $b_f(s)$20 with $b_f(s)$21 having only quasi-homogeneous isolated singularities,

$b_f(s)$22

The proof isolates the only possible new pole $b_f(s)$23, shows it cannot occur with order $b_f(s)$24, and proves vanishing of the residue by combining local-to-global arguments on the resolution graph with the non-traceless condition (Bath et al., 24 Feb 2026).

Together, these results show that in three variables the strong monodromy principle is accessible in classes where quasi-homogeneity, Newton nondegeneracy, or rigid logarithmic-derivation data sharply constrain the resolution geometry (Saito, 27 Feb 2026, Bath et al., 24 Feb 2026).

5. Tamely ramified abelian varieties

In the setting of a tamely ramified abelian variety $b_f(s)$25 of dimension $b_f(s)$26, over a Henselian discrete valuation field $b_f(s)$27 with algebraically closed residue field $b_f(s)$28, the conjecture takes a different but closely related form. Let $b_f(s)$29 be the Néron model of $b_f(s)$30, let $b_f(s)$31 in the localized Grothendieck ring $b_f(s)$32, and define

$b_f(s)$33

This motivic zeta function is rational. Writing $b_f(s)$34, the series $b_f(s)$35 has a unique pole at

$b_f(s)$36

where $b_f(s)$37 is Chai’s base-change conductor, and the order of that pole is

$b_f(s)$38

with $b_f(s)$39 the potential toric rank (Halle et al., 2010).

The strong monodromy theorem in this setting states that for every embedding $b_f(s)$40, the complex number

$b_f(s)$41

is an eigenvalue of the tame monodromy operator $b_f(s)$42 on $b_f(s)$43, and the largest Jordan block of $b_f(s)$44 with eigenvalue $b_f(s)$45 has size exactly

$b_f(s)$46

Equivalently, the pole order of the motivic zeta function equals the maximal Jordan-block size for that eigenvalue. This is the form explicitly called the strong monodromy conjecture for abelian varieties (Halle et al., 2010).

The proof proceeds through the structure of the Néron model after a finite tame extension $b_f(s)$47 giving semi-abelian reduction. The identity component of the special fiber has a Chevalley decomposition

$b_f(s)$48

and one extracts multiplicity functions $b_f(s)$49 from the Galois action on $b_f(s)$50 and $b_f(s)$51. These determine both the base-change conductor

$b_f(s)$52

and the Jordan form of tame monodromy on $b_f(s)$53. Passing to $b_f(s)$54 shifts the toric Jordan-block sizes by $b_f(s)$55, producing a block of size $b_f(s)$56 and no larger one (Halle et al., 2010).

The same paper gives a Hodge-theoretic interpretation through the limit mixed Hodge structure of a one-parameter degeneration. The graded pieces $b_f(s)$57 and $b_f(s)$58 identify with the homology of the toric and abelian parts, and the multiplicity functions appear as eigenvalue multiplicities on the Hodge-graded pieces (Halle et al., 2010).

6. Techniques, variants, and interpretive issues

Across these settings, poles are produced by numerical data $b_f(s)$59 on resolutions, or by distinguished divisors such as the exceptional divisor of the blow-up at the origin. For homogeneous degree-$b_f(s)$60 polynomials in $b_f(s)$61 variables, $b_f(s)$62 is the canonical candidate pole coming from that exceptional divisor (Bath et al., 24 Feb 2026). In hyperplane arrangements, Budur–Mustaţă–Teitler identify the smallest-stratum candidate pole as $b_f(s)$63, and other candidate poles arise from denser edges (Wu, 25 May 2026). In the meromorphic plane case, candidate poles are $b_f(s)$64 together with $b_f(s)$65 from strict-transform components of $b_f(s)$66 (Villa et al., 2013).

The proof strategies fall into several distinct but related regimes. Resolution-combinatorial arguments dominate the plane meromorphic case, where A’Campo’s formula, dual-graph analysis, and Euler-characteristic arguments on rational trees are central (Villa et al., 2013). Newton-polyhedral calculations and cancellation phenomena drive the weighted-homogeneous projective curve case, where the pole at $b_f(s)$67 is predicted by naive counting but disappears from the final expression (Saito, 27 Feb 2026). Hyperplane arrangements admit $b_f(s)$68-module and Hodge-theoretic approaches: Beilinson–Bernstein nearby cycles, proper push-forward for relative $b_f(s)$69-modules, wonderful models, propagation theorems for rank-$b_f(s)$70 local systems, and multivariate $b_f(s)$71-filtrations all enter essentially (Wu, 25 May 2026, Davis et al., 25 May 2026). For homogeneous polynomials in three variables, logarithmic derivations and residue-vanishing arguments replace purely combinatorial resolution criteria (Bath et al., 24 Feb 2026).

A recurring terminological issue is that “strong monodromy conjecture” is not completely uniform across the literature. In the hypersurface and motivic settings, the strong form usually means “pole of the zeta function implies root of the Bernstein–Sato polynomial,” with monodromy eigenvalues obtained via Malgrange–Kashiwara. Some authors additionally consider a stronger multiplicity statement comparing pole order and root multiplicity (Bath et al., 24 Feb 2026). In the abelian-variety setting, by contrast, the strong form explicitly identifies the pole order with the size of the maximal Jordan block for the corresponding eigenvalue (Halle et al., 2010).

The verified cases represented here are therefore not instances of a single proof scheme, but of a common principle realized through different structures: embedded resolutions and Milnor fibrations for meromorphic germs, intersection lattices and wonderful models for arrangements, Newton polyhedra and cancellation for plane curves in three variables, logarithmic derivations for quasi-homogeneous projective divisors, and Néron-model invariants together with limit mixed Hodge theory for abelian varieties (Villa et al., 2013, Wu, 25 May 2026, Davis et al., 25 May 2026, Saito, 27 Feb 2026, Bath et al., 24 Feb 2026, Halle et al., 2010).