Torsion points on the cohomology jump loci of compact Kähler manifolds

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.

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Accepted 23 November 2015

Published 6 June 2016