## Abstract

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.

Original language | English |
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Pages (from-to) | 39-74 |

Number of pages | 36 |

Journal | Journal fur die Reine und Angewandte Mathematik |

Issue number | 626 |

DOIs | |

State | Published - Jan 2009 |

### Bibliographical note

Funding Information:The second author was partially supported by the NSF.

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics