The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements

TL;DR

This work addresses when the root $-rac{n}{d}$ appears in the local Bernstein–Sato polynomial for indecomposable central hyperplane arrangements, a key piece of the strong topological monodromy conjecture. It develops a nonresonant extension framework on a log resolution: extend the flat connection to a logarithmic connection on $Y$, compute residues along exceptional divisors, and use vanishing theorems and the algebraic de Rham theorem to relate cohomology of a local system to a hypercohomology on $Y$. By carefully choosing residues via positive rationals and leveraging Walther’s criterion, the authors produce a nonzero class in $H^{n-1}(U,L)$ from the canonical Milnor form $oldsymbol heta_{0}$, proving that $-rac{n}{d}$ is a root of $b_{f,0}(s)$ under a nonresonant condition for weighted, indecomposable hyperplane arrangements. The results provide explicit combinatorial criteria and extend the verification of the conjecture to broader nonresonant, weighted settings, strengthening connections between hyperplane arrangement combinatorics, D-module theory, and singularity topology.

Abstract

This paper studies Bernstein--Sato polynomials $b_{f,0}$ for homogeneous polynomials $f$ of degree $d$ with $n$ variables. It is open to know when $-{n\over d}$ is a root of $b_{f,0}$. For essential indecomposable hyperplane arrangements, this is a conjecture by Budur, Mustaţă and Teitler and implies the strong topological monodromy conjecture for arrangements. Walther gave a sufficient condition that a certain differential form does not vanish in the top cohomology group of Milnor fiber. We use Walther's result to verify the $n\over d$-conjecture for weighted hyperplane arrangements satisfying the nonresonant condition.

Theorems & Definitions (3)

• proof
• proof
• proof : Proof of Theorem \ref{['thm: main']}