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Orbifold Bogomolov-Gieseker inequalities on compact Kähler varieties

Xin Fu, Wenhao Ou

Abstract

In a previous paper, the orbifold Bogomolov-Gieseker inequality is proved for a stable reflexive sheaf on a compact Kähler variety with klt singularities. In this paper, we give a characterization on the stable reflexive sheaf when the Bogomolov-Gieseker equality holds.

Orbifold Bogomolov-Gieseker inequalities on compact Kähler varieties

Abstract

In a previous paper, the orbifold Bogomolov-Gieseker inequality is proved for a stable reflexive sheaf on a compact Kähler variety with klt singularities. In this paper, we give a characterization on the stable reflexive sheaf when the Bogomolov-Gieseker equality holds.

Paper Structure

This paper contains 12 sections, 28 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex analytic varieties and complex orbifolds
  4. Local and regional fundamental groups of klt singularities
  5. Simpson's operations
  6. Uniform estimates on Kähler currents
  7. Uniform estimates on $\mathcal{W}$ classes and $\mathcal{AK}$ classes.
  8. Uniform estimates for degenerating families of orbifold Kähler forms
  9. Uniform $\mathcal{C}^0$-estimates on Hermitian-Einstein metrics
  10. Uniform mean value type inequalities for Hermitian-Einstein metrics
  11. $\mathcal{C}^0$ estimate of $h_{\epsilon,HE}$ with barrier
  12. Equality condition for Bogomolov-Gieseker inequalities

Key Result

Theorem 1.1

Let $Z$ be a projective manifold of dimension $n$, let $H$ be an ample divisor, and let ${\mathcal{F}}$ be a $H$-stable vector bundle of rank $r$ on $Z$. Then

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.4
  • ...and 50 more