Non-abelian Hodge correspondence over singular K\"ahler spaces

TL;DR

This paper extends the non-abelian Hodge correspondence to singular compact K"ahler spaces with klt singularities, establishing new equivalences and descent results, and applying them to quasi-uniformization of certain varieties.

Contribution

It generalizes the non-abelian Hodge correspondence to singular K"ahler spaces with klt singularities, including new descent theorems and applications to uniformization.

Findings

Established equivalence over regular loci via harmonic bundles

Proved descent theorem for Higgs bundles with vanishing Chern classes

Derived a quasi-uniformization theorem for specific klt varieties

Abstract

In this paper, we establish the non-abelian Hodge correspondence over compact K\"ahler spaces with Kawamata log terminal (klt) singularities as well as over their regular loci, thereby extending the result of Greb-Kebekus-Peternell-Taji for projective klt varieties to the context of compact K\"ahler klt spaces. The proof relies on two key ingredients: first, we establish an equivalence over the regular loci-via harmonic bundles-between polystable Higgs bundles with vanishing orbifold Chern numbers and semi-simple flat bundles; second, we prove a descent theorem for semistable Higgs bundles with vanishing Chern classes along resolutions of singularities. As an application of our framework, we obtain a quasi-uniformization theorem for projective klt varieties with big canonical divisor that satisfy the orbifold Miyaoka-Yau equality.