Orbifold Chern classes and Bogomolov-Gieseker inequalities
Wenhao Ou
TL;DR
This work extends Bogomolov-Gieseker-type inequalities to reflexive sheaves on compact Kähler varieties with quotient singularities in codimension 2 by developing homological orbifold Chern classes via orbifold modifications. The key method constructs orbifold vector bundles on standard orbifolds, defines $\hat{c}_1$ and $\hat{c}_2$ as linear functionals on cohomology, and proves an orbifold version of the BG inequality for $\\omega$-stable reflexive sheaves: $(2r\\hat{c}_2(\\mathcal{F}) - (r-1)\\hat{c}_1(\\mathcal{F})^2) \cdot [\\omega]^{n-2} \ge 0$. The proof leverages orbifold Hermitian-Einstein metrics on a resolved orbifold, together with a limiting process as a perturbation vanishes, and relies on recent results on orbifold modifications to relate the inequality to the original space. This result strengthens previous desingularization comparisons and supports applications to the abundance conjecture for compact Kähler threefolds.
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 Kähler form $ω$ and $\mathcal{F}$ is $ω$-stable, then we deduce Bogomolov-Gieseker inequality on the orbifold Chern classes of $\mathcal{F}$.