# Bogomolov’s inequality for Higgs sheaves in positive characteristic

We prove Bogomolov’s inequality for Higgs sheaves on varieties in positive characteristic $$p$$ that can be lifted modulo $$p^2$$. This implies the Miyaoka–Yau inequality on surfaces of non-negative Kodaira dimension liftable modulo $$p^2$$. This result is a strong version of Shepherd-Barron’s conjecture. Our inequality also gives the first algebraic proof of Bogomolov’s inequality for Higgs sheaves in characteristic zero, solving the problem posed by Narasimhan.