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-étale 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.
Highlights
1.1 Main result of this paperThe core notion of nonabelian Hodge theory, as pioneered by Corlette, Donaldson, Hitchin, and Simpson, is certainly that of a harmonic bundle
In this context, extending Simpson’s theory [37] from smooth projective manifolds to minimal models, the paper [9] established a natural equivalence between the category of local systems and the category of semistable, locally free Higgs sheaves with vanishing Chern classes on projective varieties with klt singularities
For geometric applications we need to understand locally free Higgs sheaves on the smooth locus Xreg of a klt variety X which extend to X as reflexive Higgs sheaf rather than as a locally free Higgs sheaf
Summary
The core notion of nonabelian Hodge theory, as pioneered by Corlette, Donaldson, Hitchin, and Simpson, is certainly that of a harmonic bundle. In view of the minimal model program, it is clear that these results should be studied in the more general context of varieties with terminal or canonical singularities or even for klt (= Kawamata log terminal) varieties In this context, extending Simpson’s theory [37] from smooth projective manifolds to minimal models, the paper [9] established a natural equivalence between the category of local systems and the category of semistable, locally free Higgs sheaves with vanishing Chern classes on projective varieties with klt singularities. E inherits a holomorphic structure and a Higgs field In this context, considering a compactification X of X ◦ by a simple normal crossing divisor, the notion of a tame and purely imaginary bundle (with respect to the compactification) play a decisive role. We illustrate the usefulness of Theorem 1.2 with two applications
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