Mathematical Research Letters
Volume 23 (2016)
Torsion points on the cohomology jump loci of compact Kähler manifolds
Pages: 545 – 563
We prove that each irreducible component of the cohomology jump loci of rank one local systems over a compact Kähler 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 Kähler manifolds. We also provide an example of a compact Kähler manifold, whose cohomology jump loci can not be realized by any smooth complex projective variety.