Orbifold modifications of complex analytic varieties

Abstract

We prove that if \(X\) is a compact complex analytic variety, which has quotient singularities in codimension 2, then there is a projective bimeromorphic morphism \(f\colon Y\to X\), such that \(Y\) has quotient singularities, and that the indeterminacy locus of \(f^{-1}\) has codimension at least 3 in \(X\). As an application, we deduce the Bogomolov-Gieseker inequality on orbifold Chern classes for stable reflexive coherent sheaves on compact K\"ahler varieties which have quotient singularities in codimension 2.