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

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