Deformations of Stein structures and extensions of holomorphic mappings

Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in X (and of its smooth structure when X is a Stein surface)we find a holomorphic map from X to Y which agrees with f on A and which is homotopic to f relative to A. The analogous results in the case when the variety A is empty have been obtained in the preprint math.CV/0507212.