The Monodromy Conjecture for hyperplane arrangements

Nero Budur, Mircea Mustata, Zach Teitler

Published 2009-06-10, updated 2010-01-10Version 3

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2\pi i c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every pole is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: -n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement of d hyperplanes in the affine n-space.