  We consider smooth, complex quasi-projective varieties $U$ which admit a
compactification with a boundary which is an arrangement of smooth algebraic
hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the
relative interiors of the hypersurfaces are Stein manifolds, we prove that the
cohomology of certain local systems on $U$ vanishes. As an application, we show
that complements of linear, toric, and elliptic arrangements are both duality
and abelian duality spaces.
