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 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 . These examples contradict some claims by Xie.
"The Bogomolov–Miyaoka–Yau inequality for logarithmic surfaces in positive characteristic." Duke Math. J. 165 (14) 2737 - 2769, 1 October 2016. https://doi.org/10.1215/00127094-3627203