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We extend the notion of absolute subsets of Betti moduli spaces of smoothalgebraic varieties to the case of normal varieties. As a consequence we provethat twisted cohomology jump loci in rank one over a normal variety are afinite union of translated subtori. We show that the same holds for jump locitwisted 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 theinteraction of these loci with Hodge theoretic data naturally associated to therepresentation variety of fundamental groups of smooth projective varieties.

Cohomology jump loci and absolute sets for singular varieties