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arXiv · 2104.10614

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

Yunfeng Jiang·Promit Kundu·Hao Max Sun
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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.

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Yunfeng Jiang, Promit Kundu, Hao Max Sun. 2021-04-21. On the Bogomolov-Gieseker inequality for tame Deligne-Mumford surfaces. https://arxiv.org/abs/2104.10614

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