The Miyaoka-Yau inequality for minimal Kähler klt spaces

Abstract

Let $X$ be a compact Kähler space with quotient singularities in codimension $2$. In terms of orbifold Chern classes, we prove a generalized Bogomolov-Gieseker inequality for a reflexive coherent sheaf ${\mathcal{E}}_X$ on $X$ equipped with a Higgs field ${\theta }_{X_{reg}}$ defined on the regular locus of $X$. As a direct application, we prove the Miyaoka-Yau inequality for minimal Kähler spaces. The proof relies on two main ingredients. The first is the existence of $L^p$-approximate critical Hermitian structures for orbifold Higgs bundles over compact Gauduchon orbifolds. The second is the construction of an orbifold Higgs bundle on a suitable orbifold modification $f:W\to X$, agreeing with the reflexive pullback of $({\mathcal{E}}_X,{\theta }_{X_{reg}})$ over $f^{-1}(X_{reg})$.

42 pages; v3: We fix a gap and weaken the singularity assumption in the BG inequality to "has quotient singularities in codimension 2"