Holomorphic vector bundles on holomorphically convex surface

Here we study holomorphic vector bundles on a two-dimensional holomorphically convex complex manifold Y . We present some characterizations of Y under which torsion free sheaves are filtrable. In the case when Y = X × C, where X is a connected compact Riemann surface, we study in what sense every holomorphic vector bundle may be approximated by a sequence of algebraic vector bundles.