$L^2$-Extension of Adjoint bundles and Koll\'ar's Conjecture.

We give a new proof of Kollar's conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of Albanese maps. Our technique is the $L^2$-method and we give a concrete construction and proofs of the conjecture. The $L^2$ point of view allows us to generalize Kollar's conjecture to the context of non-abelian Hodge theory.