Orbifold Chern classes and Bogomolov-Gieseker inequalities

TL;DR

This paper develops a framework for defining orbifold Chern classes on complex varieties with quotient singularities and establishes Bogomolov-Gieseker inequalities for stable sheaves using orbifold modifications.

Contribution

It introduces a method to define orbifold Chern classes on singular varieties and proves Bogomolov-Gieseker inequalities for stable sheaves in this setting.

Findings

Defined orbifold Chern classes for reflexive sheaves on quotient singularities

Established Bogomolov-Gieseker inequalities for stable sheaves with respect to a Kähler form

Extended classical inequalities to orbifold settings with singularities

Abstract

Assume that $X$ is a compact complex analytic variety which has quotient singularities in codimension 2, and that $\mathcal{F}$ is a reflexive sheaf on $X$. Using orbifold modifications, we can define first and second homological Chern classes for $\mathcal{F}$. If in addition $X$ has a K\"ahler form $\omega$ and $\mathcal{F}$ is $\omega$-stable, then we deduce Bogomolov-Gieseker inequality on the orbifold Chern classes of $\mathcal{F}$.