Harmonic metrics on Higgs sheaves and uniformization of varieties of general type
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Mathematische Annalen
Harmonic metrics on Higgs sheaves and uniformization of varieties of general type Daniel Greb1 · Stefan Kebekus2,3 · Thomas Peternell4 · Behrouz Taji5 Received: 6 April 2018 / Revised: 4 July 2019 © The Author(s) 2019
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 quasiétale 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. Mathematics Subject Classification 32Q30 · 14E20 · 14E30 · 53C07
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and elementary facts . . . . . . . . . . . . . . . . Part I. Existence of harmonic bundle structures . . . . . . . . 3 Harmonic bundles . . . . . . . . . . . . . . . . . . . . . . 4 Higgs bundles and Higgs sheaves . . . . . . . . . . . . . . 5 Existence of harmonic structures . . . . . . . . . . . . . . Part II. Applications . . . . . . . . . . . . . . . . . . . . . . . 6 Nonabelian Hodge correspondences for smooth loci . . . . 7 Proof of Theorem 1.5, uniformisation for minimal varieties 8 Positivity in the sheaf of reflexive differentials . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Communicated by Ngaiming Mok. DG was partially supported by the DFG-Collaborative Research Center SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”. SK gratefully acknowledges support through a joint fellowship of the Freiburg Institute of Advanced Studies (FRIAS) and the University of Strasbourg Institute for Advanced Study (USIAS). BT was partially supported by the DFG-Research Training Group GK1821. Research was partially completed while SK and TP were visiting the National University of Singapore in 2017. Extended author information available on the last page of the article
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1 Introduction 1.1 Main result of this paper The core notion of nonabelian Hodge theory, as pioneered by Corlette, Donaldson, Hitchin, and Simpson, is certainly that of a harmonic bundle. Most (if not all) important results of this theory depend on existence results for harmonic metrics in bundles over projective manifolds, which are usually established using highly non-trivial analytic met
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