On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces

TL;DR

This paper extends the Bogomolov-Gieseker inequality to tame Deligne-Mumford surfaces in positive characteristic, providing new tools for understanding semistable sheaves on stacks and their applications in algebraic geometry.

Contribution

It generalizes the inequality to Deligne-Mumford stacks and stacks over positive characteristic, including applications to root stacks and Simpson Higgs sheaves.

Findings

Established Bogomolov inequality for semistable sheaves on Deligne-Mumford surfaces.

Extended the inequality to characteristic zero via stack generalization.

Generalized the formula to Simpson Higgs sheaves on stacks.

Abstract

We generalize the Bogomolov-Gieseker inequality for semistable coherent sheaves on smooth projective surfaces to smooth Deligne-Mumford surfaces. We work over positive characteristic $p>0$ and generalize Langer's method to smooth Deligne-Mumford stacks. As applications we obtain the Bogomolov inequality for semistable coherent sheaves on a Deligne-Mumford surface in characteristic zero, and the Bogomolov inequality for semistable sheaves on a root stack over a smooth surface which is equivalent to the Bogomolov inequality for the rational parabolic sheaves on a smooth surface $S$. In a joint appendix with Hao Max Sun, we generalize the Bogomolov inequality formula to Simpson Higgs sheaves on tame Deligne-Mumford stacks.