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

TL;DR

This paper presents a new $L^2$-method-based proof of Kollár's conjecture on the pushforward of dualizing sheaves, extending its scope to non-abelian Hodge theory and providing concrete constructions.

Contribution

It offers a novel $L^2$-technique for Kollár's conjecture, replacing previous mixed Hodge module approaches, and generalizes the conjecture to non-abelian Hodge theory.

Findings

Provided a new proof of Kollár's conjecture using $L^2$-methods

Constructed explicit $L^2$-based proofs and generalizations

Extended the conjecture's applicability to non-abelian Hodge theory

Abstract

We give a new proof of Koll\'ar'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 Koll\'ar's conjecture to the context of non-abelian Hodge theory.