On Bogomolov’s inequality on fibered surfaces in positive characteristic

Let [Formula: see text] be a smooth projective surface over an algebraically closed field of characteristic [Formula: see text], and let [Formula: see text] be a relatively minimal fibration over a curve with smooth general fibers of genus [Formula: see text]. Let [Formula: see text] be a numerically nontrivial nef line bundle and [Formula: see text] a torsion-free coherent sheaf of rank [Formula: see text] on [Formula: see text]. We prove that if [Formula: see text] is [Formula: see text]-semistable, then [Formula: see text] where [Formula: see text]. The two choices of [Formula: see text] are deduced, respectively, from Langer’s result [8,Theorem 3.2] and Koseki’s result [7, Theorem 1.1]. As a special case, Bogomolov’s inequality holds for semistable sheaves on surfaces of product type, which was recently proved by by Sun [17].