Torsion points on the cohomology jump loci of compact Kahler manifolds

We prove that each irreducible component of the cohomology jump loci of rank one local systems over a compact Kahler manifold contains at least one torsion point. This generalizes a theorem of Simpson for smooth complex projective varieties. An immediate consequence is the positive answer to a conjecture of Beauville and Catanese for compact Kahler manifolds. We also provide an example of a compact Kahler manifold, whose cohomology jump loci can not be realized by any smooth complex projective variety.