We consider smooth, complex quasi-projective varieties that 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 the variety vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements, as well as some orbit configuration spaces of Riemann surfaces are both duality and abelian duality spaces. This is joint work with Graham Denham.