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Non-abelian Hodge correspondence over singular Kähler spaces

Chuanjing Zhang, Shiyu Zhang, Xi Zhang

Abstract

In this paper, we establish the non-abelian Hodge correspondence over compact Kähler 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 Kähler 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.

Non-abelian Hodge correspondence over singular Kähler spaces

Abstract

In this paper, we establish the non-abelian Hodge correspondence over compact Kähler 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 Kähler 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.
Paper Structure (39 sections, 60 theorems, 152 equations)

This paper contains 39 sections, 60 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications to quasi-uniformization problems
  4. Overview and main technical components
  5. Fundamental materials
  6. Complex orbifolds
  7. Higgs sheaves
  8. Higgs sheaves over the whole space
  9. Pull-back of Higgs sheaves
  10. Orbifold Chern classes
  11. Quasi-étale morphisms
  12. Pluri-harmonic metrics of Higgs sheaves over the regular loci
  13. Strategy of Theorem \ref{['main-stable']}
  14. Pull-back of the Higgs field over orbifold modifications
  15. Bando-Siu's methods
  16. ...and 24 more sections

Key Result

Theorem 1.3

Let $X$ be a compact Kähler klt space. There exists a natural one-to-one correspondence which is compatible with resolutions of singularities in the following sense. For any resolution $\pi: \widehat{X} \to X$ and any $(\mathcal{E}_X,\theta_X) \in \mathrm{Higgs}_X$, the pullback $\pi^*(\mathcal{E}_X,\theta_X)$ belongs to $\mathrm{Higgs}_{\widehat{X}}$ and the diagram \begin{tikzcd}

Theorems & Definitions (111)

  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Corollary 1.10
  • Theorem 1.11
  • ...and 101 more