Complex analytic geometry and analytic-geometric categories

The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [4]. It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets. This paper discusses connections between the subanalytic category, or more generally analytic-geometric categories, and complex analytic geometry. The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension). We then formulate conditions under which A, its closure or its image under a holomorphic map is a complex analytic set. In the second part of the paper we consider the notion of a complex ε-manifold, which generalizes that of a compact complex manifold. We discuss uniformity in parameters, in this context, within families of complex manifolds and their high-order holomorphic tangent bundles. We then prove a result on uniform embeddings of analytic subsets of ε-manifolds into a projective space, which extends theorems of Campana ([1]) and Fujiki ([6]) on compact complex manifolds.