Real Algebraic and Analytic Geometry |

Complex analytic geometry and analytic-geometric categories.

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Submission: 2005, March 14.

*Abstract:
The notion of a 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 S-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 S-manifolds into a projective space, which extends theorems
of Campana ([1]) and Fujiki ([6]) on compact complex manifolds.*

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