We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).Comment: The original publication is available at http://www.springerlink.co

Abstract. In this paper, we show that a compact real surface embedded in a complex surface has a regular Stein neighborhood basis, provided that there are only finitely many complex points on the surface, and that they are all flat and hyperbolic. An application to unions of totally real planes in C 2 is then given.

In this paper we study the structure of complex points of codimension 2 real submanifolds in complex n dimensional manifolds. We show that the local structure of a complex point up to isotopy only depends on their type (either elliptic or hyperbolic). We also show that any such submanifold can be smoothly isotoped into a submanifold that has 2-strictly pseudoconvex neighborhood basis.