L2-extension of adjoint bundles and Kollar's conjecture

We give a new proof of Kollar's conjecture on the derived push forward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture has been settled by M. Saito via the theory of mixed Hodge modules and has various applications in the investigation of Albanese maps. Our technique is L-2 theoretic which gives concrete constructions and proofs to the conjecture. The L-2 point of view allows us to generalize Kollar's conjecture to the context of non-abelian Hodge theory.