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

TL;DR

This paper develops criteria for harmonic metrics on Higgs bundles over singular varieties, leading to a uniformization result for varieties of general type and establishing a nonabelian Hodge correspondence in this setting.

Contribution

It introduces new conditions for harmonic metrics on Higgs bundles on singular varieties and applies these to solve the uniformization problem for certain algebraic varieties.

Findings

Resolved the quasi-étale uniformization problem for minimal varieties of general type.

Provided a numerical characterization of singular quotients of the unit ball.

Established a nonabelian Hodge correspondence on smooth loci of klt varieties.

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.