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

TL;DR

This paper develops a complex analytic framework for orbifold Chern classes on singular varieties, proving a Bogomolov-Gieseker inequality, characterizing equality cases, and providing new insights into complex torus quotients.

Contribution

It extends orbifold Chern class theory to singular varieties, offering new proofs and characterizations that deepen understanding of orbifold geometry and stability conditions.

Findings

Proved orbifold Bogomolov-Gieseker inequality for stable Q-sheaves

Characterized the equality case analytically

Provided a new interpretation of the second orbifold Chern class

Abstract

In this article, we prove the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves on K\"ahler 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.