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We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y . Introduction Motivated by the seminal works of Oka [40] and Grauert ([24], [25], [26]) we say that a complex manifold Y enjoys the Oka property if for every Stein manifold X, every compact O(X)-convex subset K of X and every continuous map f0 : X → Y which is holomorphic in an. | Annals of Mathematics Runge approximation on convex sets implies the Oka property By Franc Forstneri c Annals of Mathematics 163 2006 689 707 Runge approximation on convex sets implies the Oka property By Franc Forstneric Abstract We prove that the classical Oka property of a complex manifold Y concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y is equivalent to a Runge approximation property for holomorphic maps from compact convex sets in Euclidean spaces to Y. Introduction Motivated by the seminal works of Oka 40 and Grauert 24 25 26 we say that a complex manifold Y enjoys the Oka property if for every Stein manifold X every compact O X -convex subset K of X and every continuous map f0 X - Y which is holomorphic in an open neighborhood of K there exists a homotopy of continuous maps ft X Y t E 0 1 such that for every t E 0 1 the map ft is holomorphic in a neighborhood of K and uniformly close to f0 on K and the map fl X - Y is holomorphic. The Oka property and its generalizations play a central role in analytic and geometric problems on Stein manifolds and the ensuing results are commonly referred to as the Oka principle. Applications include the homotopy classification of holomorphic fiber bundles with complex homogeneous fibers the Oka-Grauert principle 26 7 31 and optimal immersion and embedding theorems for Stein manifolds 9 43 for further references see the surveys 15 and 39 . In this paper we show that the Oka property is equivalent to a Runge-type approximation property for holomorphic mappings from Euclidean spaces. Theorem 0.1. If Y is a complex manifold such that any holomorphic map from a neighborhood of a compact convex set K c Cn n E N to Y can be approximated uniformly on K by entire maps Cn Y then Y satisfies the Oka property. Research supported by grants P1-0291 and J1-6173 Republic of Slovenia. 690 FRANC FORSTNERIC The hypothesis in Theorem 0.1 will be referred to as the convex approximation .
