Real Algebraic and Analytic Geometry

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139. Ya'acov Peterzil, Sergei Starchenko:
Complex analytic geometry and analytic-geometric categories.

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Submission: 2005, March 14.

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.

Full text, 33p.: dvi 159k, ps.gz 235k, pdf 326k.

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