The Jacobian module, the Milnor fiber, and the $D$-module generated by $f^s$

TL;DR

This paper introduces a complex linking the Jacobian module, Milnor fiber, and D-module generated by f^s, establishing new connections and results for hyperplane arrangements and their Bernstein--Sato polynomials.

Contribution

It develops a Liouville complex to relate D-module properties to Jacobian ideals, proving conjectures for hyperplane arrangements and analyzing the Bernstein--Sato polynomial.

Findings

Proves Terao's conjecture on the annihilator of 1/f for arrangements.

Shows the Bernstein--Sato polynomial is not solely determined by the intersection lattice.

Arrangements with derivation-generated annihilators satisfy the Strong Monodromy Conjecture.

Abstract

For a germ $f$ on a complex manifold $X$, we introduce a complex derived from the Liouville form acting on logarithmic differential forms, and give an exactness criterion. We use this Liouville complex to connect properties of the $D$-module generated by $f^s$ to homological data of the Jacobian ideal; specifically we show that for a large class of germs the annihilator of $f^s$ is generated by derivations. Through local cohomology, we connect the cohomology of the Milnor fiber to the Jacobian module via logarithmic differentials. In particular, we consider (not necessarily reduced) hyperplane arrangements: we prove a conjecture of Terao on the annihilator of $1/f$; we confirm in many cases a corresponding conjecture on the annihilator of $f^s$ but we disprove it in general; we show that the Bernstein--Sato polynomial of an arrangement is not determined by its intersection lattice; we prove that arrangements for which the annihilator of $f^s$ is generated by derivations fulfill the Strong Monodromy Conjecture, and that this includes as very special cases all arrangements of Coxeter and of crystallographic type, and all multi-arrangements in dimension 3.