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Extending holomorphic mappings from subvarieties in Stein manifolds. (English) Zbl 1076.32003

A complex manifold \(Y\) satisfies the convex approximation property (CAP) if every holomorphic map \(f: U\to Y\) from an open set \(U\subset{\mathbb C}^n\) (\(n\in{\mathbb N}\)) can be approximated uniformly on any compact convex set \(K\subset U\) by entire maps \({\mathbb C}\to Y\).
The main result of this paper is that CAP implies the universal extendability of holomorphic maps from closed complex subvarieties in Stein manifolds. Assume that \(Y\) is a complex manifold \(Y\) satisfying CAP. For any closed complex subvariety \(X_0\) in Stein manifold \(X\) and any continuous map \(f_0: X\to Y\) such that \(f_0| _{X_0}:X_0\to Y\) is holomorphic there is a homotopy \(f_t: X\to Y\) (\(t\in[0,1]\)) which is fixed on \(X_0\) such that \(f_1\) is holomorphic on \(X\). The analogous conclusion holds for sections of any holomorphic fiber bundle with fiber \(Y\) over a Stein manifold. Summarizing the results of this paper the author establishes the equivalence of four Oka-type properties of a complex manifold.

MSC:

32E10 Stein spaces
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
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