The Oka principle for multivalued sections of ramified mappings

We prove a basic version of the Oka principle for multivalued sections of ramified holomorphic mappings h : Z --> X of a complex space Z onto an irreducible Stein space X. Assume that X-0 subset of X is a closed complex subvariety of X such that h is an elliptic submersion over X\X-0, in the sense that it admits a fiber-dominating spray over a small neighborhood of any point in X\X-0. If F-0 is a d-valued continuous section of h which is holomorphic in a neighborhood of X-0, unramified over X\X-0 and whose discriminant locus has Hausdorff (2n - 1)-dimensional measure zero, where n = dim X, then F-0 can be homotopically deformed to a holomorphic d-valued section F of h. Since the graph V (F) of such F is an effective analytic chain in Z such that h : V(F) --> X is an analytic cover, we obtain an existence result for such chains. Our results apply to maps h whose generic fibers are parabolic or elliptic curves.