  The topology of smooth quasi-projective complex varieties is very
restrictive. One aspect of this statement is the fact that natural strata of
local systems, called cohomology support loci, have a rigid structure: they
consist of torsion-translated subtori in a complex torus. We propose and
partially confirm a relation between Bernstein-Sato ideals and local systems.
This relation gives yet a different point of view on the nature of the
structure of cohomology support loci of local systems. The main result is a
partial generalization to the case of a collection of polynomials of the
theorem of Malgrange and Kashiwara which states that the Bernstein-Sato
polynomial of a hypersurface recovers the monodromy eigenvalues of the Milnor
fibers of the hypersurface. We also address a multi-variable version of the
Monodromy Conjecture, prove that it follows from the usual single-variable
Monodromy Conjecture, and prove it in the case of hyperplane arrangements.
