The linear Shafarevich conjecture for quasiprojective varieties and algebraicity of Shafarevich morphisms

We prove that the universal cover of a normal complex algebraic variety admitting a faithful complex representation of its fundamental group is an analytic Zariski open subset of a holomorphically convex complex space. This is a non-proper version of the Shafarevich conjecture. More generally we define a class of subset of the Betti stack for which the covering space trivializing the corresponding local systems has this property. Secondly, we show that for any complex local system V on a normal complex algebraic variety X there is an algebraic map f X Y contracting precisely the subvarieties on which V is isotrivial.