Complex analytic geometry and analytic-geometric categories

The notion of an analytic-geometric category was introduced by v. d. Dries and Miller in [Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540.]. 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 -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 -manifolds into a projective space, which extends theorems of Campana ([F. Campana, Algébricité et compacité dans l'espace des cycles d'un espace analytique complexe, Math. Ann. 251 (1980), no. 1, 7–18.]) and Fujiki ([Akira Fujiki, On the Douady space of a compact complex space in the category C, Nagoya Math. J. 85 (1982), 189–211.]) on compact complex manifolds.