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In this paper we establish a basic version of the Oka principle formultivalued sections of ramified holomorphic maps h from a complex manifold Zonto a Stein manifold X. If the ramification locus of h projects into a closedcomplex subvariety X' of X and if h admits a fiber dominating spray over asmall neighborhood of any point in X\X' then any multivalued continuous sectionof h which is holomorphic in a neighborhood of X' and unramified in X\X' can behomotopically deformed to a global holomorphic multivalued section. Thecorresponding results for sections of unramified submersions were establishedby Gromov (Oka's principle for holomorphic sections of elliptic bundles, J.Amer. Math. Soc. 2, 851-897 (1989)) and Forstneric and Prezelj (Math. Ann., toappear).

The Oka principle for multivalued sections of ramified mappings