Hodge modules on complex tori and generic vanishing for compact K\"ahler manifolds

We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact K"ahler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori among compact K"ahler manifolds, semi-positivity results, and a description of the Leray filtration for maps to tori.