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