$$L^2$$-extension of adjoint bundles and Kollár’s conjecture

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