Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles

Abstract

A. We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. e proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singularities, then any vector bundle on the resolution that appears to come from X numerically, does indeed come from X . Furthermore and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.We refer the reader to [GKPT15, Sect. 5] or to the survey [GKT18, Sect. 6] for the (rather delicate) notions of Higgs sheaves, morphisms of Higgs sheaves and pull-back. Semistability is also discussed there.

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

“…A morphism of vector bundles is always assumed to have constant rank. Notation introduced in our previous papers, [ GKPT15,GKPT19], will briefly be recalled before it is used.…”

“…In direct analogy to Simpson's work, eorem 6.3 extends to give an equivalence between the category of flat bundles and arbitrary local systems on X reg . e (fairly standard) proof requires a version of the nonabelian Hodge correspondence for a maximally quasi-étale cover, [GKPT19,m. 3.4 and the discussion a er Prop.…”

“…Assigning (E X reg , θ X reg ) to (E X reg , ∇ E Xreg ) defines a functor η X reg from sLSys X reg to pHiggs X reg . Compatibility with the global nonabelian Hodge Correspondence for projective klt spaces follows from the construction in [GKPT19]; cf. especially [GKPT19,Prop.…”

“…Compatibility with the global nonabelian Hodge Correspondence for projective klt spaces follows from the construction in [GKPT19]; cf. especially [GKPT19,Prop. 3.10].…”

“…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 [Sim92] from smooth projective manifolds to minimal models, the paper [GKPT19] 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 eorem 1.1 ([GKPT19, m. 1.1]). Let X be a projective, klt variety.…”