Bernstein–Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements

Abstract

Let $Q\in{\mathbb C}[x_1,\dotsc,x_n]$ be a homogeneous polynomial of degree $k>0$. We establish a connection between the Bernstein–Sato polynomial $b_Q(s)$ and the degrees of the generators for the top cohomology of the associated Milnor fiber. In particular, the integer $u_Q={\rm max}\{i\in{\mathbb Z}:b_Q(-(i+n)/k)=0\}$ bounds the top degree (as differential form) of the elements in $H^{n-1}_{\rm DR}(Q^{-1}(1),{\mathbb C})$. The link is provided by the relative de Rham complex and ${\mathcal D}$-module algorithms for computing integration functors.

As an application we determine the Bernstein–Sato polynomial $b_Q(s)$ of a generic central arrangement $Q=\prod_{i=1}^kH_i$ of hyperplanes. In turn, we obtain information about the cohomology of the Milnor fiber of such arrangements related to results of Orlik and Randell who investigated the monodromy.

We also introduce certain subschemes of the arrangement determined by the roots of $b_Q(s)$. They appear to correspond to iterated singular loci.