Cohomology bounds and Chern class inequalities for stable sheaves on a smooth projective variety

We give effective upper bounds for dimensions of the (n - 1)-th cohomology groups of mu-semistable torsion-free sheaves on a smooth projective variety of dimension n defined over an algebraically closed fieled of characteristic zero. As a corollary to this result, we obtain bounds for the dimension of the moduli space of mu-stable vector bundles. We also prove Bogomolov-Gieseker type inequalities for the fourth Chern classes c(4)(E) of mu-semistable vector bundles E on a smooth projective fourfold.