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Bogomolov-Gieseker inequality for log terminal Kähler threefolds

Henri Guenancia, Mihai Păun

TL;DR

The paper proves an orbifold Bogomolov-Gieseker inequality for stable reflexive $\mathbb{Q}$-sheaves on 3-dimensional compact Kähler spaces with log terminal singularities: for such a sheaf $\mathcal{F}$, the discriminant current satisfies $[\omega_X]$-intersection nonnegativity, i.e. $\\Delta({\mathcal F})\\cdot [\omega_X]\\ge 0$. The authors develop analytic tools—canonical singular metrics $\omega$ solving a Monge-Ampère equation, interpolating metrics $\omega_\theta$, and Hermite-Einstein metrics $h_{\mathcal F}$—and leverage Green's function estimates from GPSS to control currents and boundary terms. The equality case yields that ${\mathcal F}$ is projectively flat on the smooth locus and extends, after a finite quasi-étale cover, to a locally trivial $\mathbb P^{r-1}$-bundle, supporting a conjecture of Campana–Höring–Peternell and informing abundance-type questions for Kähler threefolds. Overall, the work provides a robust analytic framework for BG-type inequalities on singular Kähler spaces and connects orbifold Chern-class concepts to stability theory, with implications for the structure of vector bundles and abundance results in complex geometry.

Abstract

In this article we are mainly concerned with three dimensional compact Kähler spaces with log terminal singularities. We establish the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves.

Bogomolov-Gieseker inequality for log terminal Kähler threefolds

TL;DR

The paper proves an orbifold Bogomolov-Gieseker inequality for stable reflexive -sheaves on 3-dimensional compact Kähler spaces with log terminal singularities: for such a sheaf , the discriminant current satisfies -intersection nonnegativity, i.e. . The authors develop analytic tools—canonical singular metrics solving a Monge-Ampère equation, interpolating metrics , and Hermite-Einstein metrics —and leverage Green's function estimates from GPSS to control currents and boundary terms. The equality case yields that is projectively flat on the smooth locus and extends, after a finite quasi-étale cover, to a locally trivial -bundle, supporting a conjecture of Campana–Höring–Peternell and informing abundance-type questions for Kähler threefolds. Overall, the work provides a robust analytic framework for BG-type inequalities on singular Kähler spaces and connects orbifold Chern-class concepts to stability theory, with implications for the structure of vector bundles and abundance results in complex geometry.

Abstract

In this article we are mainly concerned with three dimensional compact Kähler spaces with log terminal singularities. We establish the orbifold version of the Bogomolov-Gieseker inequality for stable -sheaves.
Paper Structure (16 sections, 24 theorems, 161 equations)

This paper contains 16 sections, 24 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Bogomolov-Gieseker inequality
  3. Canonical metrics on Kähler spaces with log terminal singularities
  4. Canonical metrics on log terminal Kähler spaces
  5. Construction and properties of Green's functions
  6. Some properties of canonical metrics
  7. Orbifold regularity
  8. A direct construction
  9. The Bogomolov-Gieseker inequality
  10. Geometric setup and statement of the main result
  11. Construction of an interpolating metric and its main properties
  12. The induced Hermite-Einstein metric on ${\mathcal{F}}$
  13. $C^0$ estimates on the orbifold locus
  14. $\overline{\partial}$-closedness of the current induced by $\Delta({\mathcal{F}}, h_{\mathcal{F}})$ on the orbifold locus
  15. Proof of Theorem \ref{['thm1']}
  16. ...and 1 more sections

Key Result

Theorem A

Let $(X,\omega_X)$ be a compact Kähler variety of dimension three with log terminal singularities. Let ${\mathcal{F}}$ be a reflexive $\mathbb Q$-sheaf on $X$ which is stable with respect to $[\omega_X]$. Then the Bogomolov-Gieseker inequality holds for ${\mathcal{F}}$, i.e. In the equality case, ${\mathcal{F}}|_{X_{\rm reg}}$ is a projectively flat vector bundle.

Theorems & Definitions (54)

  • Theorem A
  • Proposition B
  • Definition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3: $L^{\infty}$-estimate
  • proof
  • Remark 3.4
  • Remark 3.5
  • ...and 44 more