  We extend the notion of absolute subsets of Betti moduli spaces of smooth
algebraic varieties to the case of normal varieties. As a consequence we prove
that twisted cohomology jump loci in rank one over a normal variety are a
finite union of translated subtori. We show that the same holds for jump loci
twisted by a unitary local system in the case where the underlying variety $X$
is projective with $H^1(X,\mathbb{Q})$ pure of weight one. Lastly, we study the
interaction of these loci with Hodge theoretic data naturally associated to the
representation variety of fundamental groups of smooth projective varieties.
