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