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A complex analytic approach to orbifold Chern classes on singular varieties and its applications

Henri Guenancia, Mihai Păun

TL;DR

The paper extends the Bogomolov-Gieseker framework to singular Kähler spaces by proving a singular BG inequality for stable ${\mathbb Q}$-sheaves and characterizing the equality case as projective flatness on the regular locus. It develops a cohesive analytic framework to define and compute orbifold Chern numbers via Dolbeault and de Rham cohomology, using a decomposition $\omega^{n-2}=\Omega+\overline{\partial}\gamma$ and Leray-type maps to connect coherent, Dolbeault, and de Rham perspectives. These tools enable a purely analytic route to uniformization-type results, including a numerical criterion for torus quotients in the singular setting, and provide a robust methodology for interpreting orbifold Chern data on singular varieties. The combination of analytic BG-type estimates, global closedness of curvature, and cohomological orbifold formulations paves the way for applications to the geometry of Kähler spaces with quotient singularities and their uniformization properties.

Abstract

In this article, we prove the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves on Kähler varieties, generalizing our earlier work \cite{GP25} in dimension three. We also provide a characterization of the equality case, a new purely analytical proof of the numerical characterization of complex torus quotients as well as a novel, complex analytic interpretation of the second orbifold Chern class associated to a $\mathbb Q$-sheaf.

A complex analytic approach to orbifold Chern classes on singular varieties and its applications

TL;DR

The paper extends the Bogomolov-Gieseker framework to singular Kähler spaces by proving a singular BG inequality for stable -sheaves and characterizing the equality case as projective flatness on the regular locus. It develops a cohesive analytic framework to define and compute orbifold Chern numbers via Dolbeault and de Rham cohomology, using a decomposition and Leray-type maps to connect coherent, Dolbeault, and de Rham perspectives. These tools enable a purely analytic route to uniformization-type results, including a numerical criterion for torus quotients in the singular setting, and provide a robust methodology for interpreting orbifold Chern data on singular varieties. The combination of analytic BG-type estimates, global closedness of curvature, and cohomological orbifold formulations paves the way for applications to the geometry of Kähler spaces with quotient singularities and their uniformization properties.

Abstract

In this article, we prove the orbifold version of the Bogomolov-Gieseker inequality for stable -sheaves on Kähler varieties, generalizing our earlier work \cite{GP25} in dimension three. We also provide a characterization of the equality case, a new purely analytical proof of the numerical characterization of complex torus quotients as well as a novel, complex analytic interpretation of the second orbifold Chern class associated to a -sheaf.
Paper Structure (21 sections, 17 theorems, 90 equations)

This paper contains 21 sections, 17 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Orbifold Chern numbers : how to compute them
  4. What goes on in the proof of Theorem \ref{['thmA']}
  5. Notations and setup
  6. Recollection from GP25$+\varepsilon$
  7. Cutoff functions and a Harnack-type result
  8. $C^0$ estimates for $h_{{\mathcal{F}}}$ on $X\setminus Z$
  9. End of the proof: integration by parts on $X\setminus Z$
  10. Closedness of $\Theta$ on $X$
  11. Orbifold Chern classes revisited
  12. The Leray map
  13. From coherent to Dolbeault cohomology
  14. From singular to de Rham cohomology
  15. Orbifold Chern numbers via de Rham cohomology
  16. ...and 6 more sections

Key Result

Theorem A

Let $(X,\omega)$ be a compact Kähler space of dimension $n$ and let ${\mathcal{F}}$ be a coherent reflexive sheaf on $X$ of rank $r$ which is $[\omega]$-stable. Assume that there exists a closed analytic subset $Z\subset X$ of codimension at least three such that $X\setminus Z$ has at most quotient and equality holds if and only if ${\mathcal{F}}|_{X_{\rm reg}}$ is locally free and projectively f

Theorems & Definitions (29)

  • Theorem A
  • Corollary B
  • Theorem C
  • Remark 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Proposition 3.5
  • Theorem 4.1
  • proof
  • ...and 19 more