The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic

Abstract

We generalize Bogomolov’s inequality for Higgs sheaves and the Bogomolov– Miyaoka–Yau inequality in positive characteristic to the logarithmic case. We also generalize Shepherd-Barron’s results on Bogomolov’s inequality on surfaces of special type from rank $2$ to the higher-rank case. We use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo $p^2$. These examples contradict some claims by Xie.