Slope inequalities and a Miyaoka–Yau type inequality

Abstract

For a minimal smooth projective surface $S$ of general type over a field of characteristic $p>0$, we prove that

Moreover, if $18\chi ({\mathcal{O}}_S)<K_S^2\le 32\chi ({\mathcal{O}}_S)$, the Albanese morphism of $S$ must induce a genus 2 fibration. A classification of surfaces with $K_S^2=32\chi ({\mathcal{O}}_S)$ is also given. The inequality also implies $\chi ({\mathcal{O}}_S)>0$, which answers completely a question of Shepherd-Barron.