Bogomolov type inequalities and Frobenius semipositivity

We prove Bogomolov type inequalities for high Chern characters of semistable sheaves satisfying certain Frobenius semipositivity. The key ingredients in the proof are a high rank generalization of the asymptotic Riemann-Roch theorem and Langer’s estimation theorem of the global sections of torsion free sheaves. These results give some Bogomolov type inequalities for semistable sheaves with vanishing low Chern characters. Our results are also applied to obtain inequalities of Chern characters of threefolds and varieties of small codimension in projective spaces and abelian varieties.