  We give two applications of the exponential Ax-Lindemann Theorem to local
systems. One application is to show that for a connected topological space, the
existence of a finite model of the real homotopy type implies linearity of the
cohomology jump loci around the trivial local system. Another application is
the linearity of the cohomology jump loci of rank one local systems on
quasi-compact K\"ahler manifolds.
