The strong monodromy conjecture for hyperplane arrangements

Abstract: In this article, we prove the strong monodromy conjecture for complex hyperplane arrangements by proving a conjecture of Budur, Musta\c t\u a and Teitler that $-n/d$ is a root of the $b$-function of an irreducible essential and central hyperplane arrangement $f$ of degree $d$ on $\C^n$.

• The paper proves that for any irreducible, essential, central hyperplane arrangement, every -k/d (for k = n,...,d) is a root of the Bernstein–Sato polynomial b_f(s).
• It employs advanced algebraic D-module techniques and explicit resolution methods, bypassing traditional Hodge module theory, to establish the conjecture.
• The approach generalizes to Bernstein–Sato ideals and offers new insights into singularity invariants and the geometry of hyperplane arrangements.

The Strong Monodromy Conjecture for Hyperplane Arrangements

Background and Motivation

The monodromy conjecture, originating with Denef and Loeser, links the poles of local zeta functions associated to hypersurface singularities with two independent structures: the eigenvalues of the Milnor monodromy (topological monodromy) and the roots of the Bernstein–Sato polynomial ($b$-function) of a polynomial $f$. The conjecture is widely regarded as a fundamental and deep question in singularity theory, with ramifications in D-module theory, Hodge theory, and algebraic geometry.

The conjecture states, in its strong form, that if $c$ is a pole of the topological zeta function of $f$, then $c$ is a root of the $b$-function $b_f(s)$. For hyperplane arrangements, the conjecture has received special attention: Budur, Mustață, and Teitler [BMT] established the weak form for this case and connected the strong form to the vanishing of $b_f(-n/d)$, where $f$ defines a central, essential, irreducible arrangement in $\mathbb{C}^n$ of degree $f$0.

Conjecture [BMT] posits that for such $f$1, $f$2. This central conjecture has been resolved only in important special cases prior to this work, such as for $f$3, Weyl arrangements, tame arrangements, and arrangements with generic multiplicities.

Main Results

The paper establishes the strong monodromy conjecture for all complex hyperplane arrangements (2605.25335). Precisely, it proves that for any irreducible, essential, and central hyperplane arrangement $f$4 of degree $f$5 on $f$6, $f$7 is a root of $f$8 for $f$9, thereby confirming the Budur–Mustață–Teitler conjecture in full generality. The methods do not rely on Hodge module theory, in contrast to preceding claims in the literature.

The approach generalizes to Bernstein–Sato ideals for such arrangements, capturing broader phenomena than the case of reduced arrangements.

Technical Approach

The central argument employs advanced techniques from the theory of algebraic D-modules, particularly the machinery of relative holonomic D-modules. The proof leverages the following key advances:

• Commutativity of duality and direct images for relative D-modules: The author establishes that duality functors commute with proper pushforwards in the algebraic (not just analytic) category. This is crucial for transporting local properties of D-modules across resolutions.
• Beilinson–Bernstein construction of D-module nearby cycles: Utilizing right-module versions of nearby cycles, the analysis translates subtle properties of the $c$0-function roots to properties of these functors.
• Wonderful model of De Concini–Procesi: This canonical log resolution for arrangements is essential for uniform control over exceptional divisors and understanding the subtle combinatorics and geometry of hyperplane arrangements at the level of D-modules.
• Special direct images and explicit calculation of sections 'of the form $c$1': By identifying the image of these sections under the resolution and using properties of the arrangement complement, the necessary non-vanishing result is secured.
• Propagation property of cohomology jumping loci: This recent property (from [DSYprop]) ensures non-vanishing in the appropriate cohomology of rank-one local systems, underpinning the proof of non-triviality for the relevant D-module sections.

A notable aspect of the work is an alternative proof of the classical Kashiwara–Malgrange rationality theorem for $c$2-functions and Lichtin's explicit description of their roots, done purely within relative D-module theory and the Beilinson–Bernstein approach.

Numerical and Structural Results

The main theorem asserts, in unambiguous terms, that the set $c$3 consists of roots of $c$4 for any irreducible, essential, central hyperplane arrangement $c$5 of degree $c$6 in $c$7. This is both stronger and more explicit than prior results; for instance, it implies Budur–Mustață–Teitler's conjecture and more, by directly identifying an infinite family of roots based on the fundamental parameters of the arrangement.

Implications and Future Directions

The theoretical implications of this proof are multifaceted:

• It settles a long-standing conjecture, solidifying the conceptual link between singularity invariants associated to hyperplane arrangements.
• The techniques employed indicate uniformity and flexibility in the algebraic theory of D-modules, particularly in settings (such as arrangements) where the geometry is both highly combinatorial and analytic.
• The extension to Bernstein–Sato ideals hints at similar approaches being applicable to more general or non-reduced settings, possibly informing studies in other classes of free divisors or multiarrangements.
• By eschewing Hodge-theoretic arguments in favor of D-module methods and explicit resolution geometry, the work broadens the arsenal for attacks on other variants of the monodromy conjecture.

These developments suggest new avenues in the study of invariants of algebraic varieties, especially in exploiting D-module methods for explicit calculations and deeper theoretical insights.

Conclusion

By proving that $c$8 (and, in fact, all $c$9 for $f$0) are roots of the Bernstein–Sato polynomial for any irreducible, essential, central hyperplane arrangement of degree $f$1 in $f$2, this work unconditionally establishes the strong monodromy conjecture for hyperplane arrangements. The methods illustrate the power and generality of algebraic D-module theory coupled with explicit resolution models in algebraic geometry, and their impact is expected to influence future research into singularities, free divisors, and beyond (2605.25335).