Deformations of Stein structures and extensions of holomorphic mappings

Assume that $A$ is a closed complex subvariety of a Stein manifold $X$ and that $f\colon X\to Y$ is a continuous map to a complex manifold $Y$ such that the restriction $f|_A\colon A\to Y$ is holomorphic on $A$. After a homotopic deformation of the Stein structure outside a neighborhood of $A$ in $X$ we find a holomorphic map $\wt f\colon X\to Y$ which agrees with $f$ on $A$ and which is homotopic to $f$ relative to $A$. When $\dim_\C X=2$ we must also change the $\cC^\infty$ structure on $X\bs A$.

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Published 1 January 2007