  We prove the Bogomolov-Gieseker type inequality conjectured by Bayer, Macri
and Toda for threefolds with semistable tangent bundles and vanishing Chern
classes in any characteristic, which was originally proved by Bayer, Macri and
Stellari in characteristic zero. This gives the existence of Bridgeland
stability conditions on such threefolds. As applications, we obtain Reider type
theorem and confirm Fujita's conjecture for such threefolds in any
characteristic.
