On the Bogomolov-Gieseker inequality for hypersurfaces in the projective spaces

We prove a version of the Bogomolov-Gieseker inequality on smooth projective surfaces of general type in positive characteristic, which is stronger than the result by Langer when the ranks of vector bundles are sufficiently large. Our inequality enables us to construct Bridgeland stability conditions with full support property on all smooth projective surfaces in positive characteristic.