Harmonic metrics on Higgs sheaves and uniformization of varieties of general type

Abstract

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one. As a further application, we establish a nonabelian Hodge correspondence on smooth loci of klt varieties.