Oka manifolds

We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact convex set in a complex Euclidean space C^n to a certain complex manifold Y is a uniform limit of entire maps of C^n to Y, then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion of the space of holomorphic maps of X to Y into the space of continuous maps is a weak homotopy equivalence. This shows that all Oka type properties of a complex manifold are equivalent to each other. (See also the articles F. Forstneric, "Runge approximation on convex sets implies Oka's property", Ann. Math. (2), 163, 689-707 (2006); "Extending holomorphic mappings from subvarieties in Stein manifolds", Ann. Inst. Fourier 55, 733-751 (2005).)

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