First order deformations for rank one local systems with a non-vanishing cohomology

We study the tangent cone to the variety Vkm of rank one local systems on a finite CW-complex for which the dimension of k-dimensional cohomology is at least m. This cone is related to the space of certain complexes of abelian groups with differential induced by the cup product. In particular, for an arrangement of hyperplanes, the set of 1-forms for which the corresponding Aomoto complex has k-dimensional Betti number greater or equal to m is a union of linear spaces.