Hodge modules on complex tori and generic vanishing for compact Kähler manifolds

Abstract

We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact Kähler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori (among compact Kähler manifolds), semipositivity results and a description of the Leray filtration for maps to tori.