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The Miyaoka-Yau inequality for minimal Kähler klt spaces

Chuanjing Zhang, Shiyu Zhang, Xi Zhang

TL;DR

This work establishes a generalized Bogomolov–Gieseker type inequality for reflexive Higgs sheaves on the regular locus of compact Kähler spaces with klt singularities and derives the Miyaoka–Yau inequality for all minimal Kähler klt spaces. The authors develop an $L^p$-approximate Hermitian–Einstein theory for Higgs orbi-bundles on Gauduchon orbifolds, and use orbifold Chern classes to translate singular invariants into tractable curvature expressions. A comprehensive framework for Higgs sheaves on the regular locus is built, including stability, Harder–Narasimhan filtrations, and pull-back calculus, which then feeds into global positivity results via partial orbifold resolutions. The main contributions provide a self-contained Higgs-theoretic treatment on complex normal spaces and furnish tools to prove positivity of orbifold Chern classes and semistability properties in singular settings, enabling the orbifold Miyaoka–Yau bound in the Kähler klt context. Collectively, the paper extends fundamental geometric inequalities from smooth to singular analytic spaces and offers analytic and algebro-geometric machinery applicable to stability, Chern-class calculus, and curvature methods in complex geometry with singularities.

Abstract

In this paper, we obtain the generalized Bogomolov inequality for reflexive Higgs sheaves defined on the regular locus of compact Kähler klt spaces. As an application, we establish the Miyaoka-Yau inequality for all minimal Kähler klt spaces. Apart from providing a self-contained formulation and investigation of Higgs sheaves on complex normal spaces, the analytical part of our approach is the establishment of $L^p$-approximate critical Hermitian structures for Higgs orbi-bundles on Gauduchon orbifolds. This also leads to the semistability (resp. generically nefness) of torsion-free sheaves under symmetric, exterior powers and tensor products in the singular setting.

The Miyaoka-Yau inequality for minimal Kähler klt spaces

TL;DR

This work establishes a generalized Bogomolov–Gieseker type inequality for reflexive Higgs sheaves on the regular locus of compact Kähler spaces with klt singularities and derives the Miyaoka–Yau inequality for all minimal Kähler klt spaces. The authors develop an -approximate Hermitian–Einstein theory for Higgs orbi-bundles on Gauduchon orbifolds, and use orbifold Chern classes to translate singular invariants into tractable curvature expressions. A comprehensive framework for Higgs sheaves on the regular locus is built, including stability, Harder–Narasimhan filtrations, and pull-back calculus, which then feeds into global positivity results via partial orbifold resolutions. The main contributions provide a self-contained Higgs-theoretic treatment on complex normal spaces and furnish tools to prove positivity of orbifold Chern classes and semistability properties in singular settings, enabling the orbifold Miyaoka–Yau bound in the Kähler klt context. Collectively, the paper extends fundamental geometric inequalities from smooth to singular analytic spaces and offers analytic and algebro-geometric machinery applicable to stability, Chern-class calculus, and curvature methods in complex geometry with singularities.

Abstract

In this paper, we obtain the generalized Bogomolov inequality for reflexive Higgs sheaves defined on the regular locus of compact Kähler klt spaces. As an application, we establish the Miyaoka-Yau inequality for all minimal Kähler klt spaces. Apart from providing a self-contained formulation and investigation of Higgs sheaves on complex normal spaces, the analytical part of our approach is the establishment of -approximate critical Hermitian structures for Higgs orbi-bundles on Gauduchon orbifolds. This also leads to the semistability (resp. generically nefness) of torsion-free sheaves under symmetric, exterior powers and tensor products in the singular setting.

Paper Structure

This paper contains 27 sections, 51 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Fundamental materials
  3. $\mathbb{Q}$-line bundle and Positivity
  4. Complex orbifolds
  5. Chern classes of orbi-sheaves
  6. Chern-Weil formula
  7. HN filtration of Higgs orbi-sheaves
  8. Homology Chern classes
  9. The homology first Chern class
  10. Push-forward of orbi-sheaves to the quotient space
  11. Orbifold Chern classes
  12. $L^p$-approximate critical Hermitian structure
  13. Main results
  14. Proof of Theorem \ref{['thm1']} and Theorem \ref{['thm2']}
  15. Higgs sheaves on the regular locus
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $X$ be a compact Kähler space of dimension $n$ with klt singularities and nef canonical sheaf. Then for any Kähler form $\omega_X$ on $X$, we have where $i=\min(v(K_X),n-2)$, $\widehat{c}_2(X)=\widehat{c}_2(\mathcal{T}_X)$ and $\widehat{c}_1(X)^2=\widehat{c}_1(\mathcal{T}_X)^2$ denotes Orbifold Chern classes of $X$.

Theorems & Definitions (104)

  • Theorem 1.1: Orbifold Miyaoka-Yau inequality
  • Theorem 1.2: Generalized Bogomolov-type inequality
  • Corollary 1.3
  • Corollary 1.4
  • Definition 2.1: $\mathbb{Q}$-line bundle
  • Definition 2.2: $\mathbb{Q}$-Gorenstein spaces
  • Definition 2.3: BG13
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6: Kähler forms
  • ...and 94 more