Approximation of semistable bundles on smooth algebraic varieties

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran-Subramanian in the curve case and Koley-Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.