Stein neighborhood bases of embedded strongly pseudoconvex domains and approximation of mappings

In this paper we construct a Stein neighborhood basis for any compact subvariety $A$ with strongly pseudoconvex boundary $bA$ and Stein interior $A\mathrm{\backslash }bA$ in a complex space $X$. This is an extension of a well known theorem of Siu. When $A$ is a complex curve, our result coincides with the result proved by Drinovec-Drnovšek and Forstnerič. We shall adapt their proof to the higher dimensional case, using also some ideas of Demailly’s proof of Siu’s theorem. For embedded strongly pseudoconvex domain in a complex manifold we also find a basis of tubular Stein neighborhoods. These results are applied to the approximation problem for holomorphic mappings.