Submission date: 8 January 2008. Revised version, 29 January 2008.

Abstract:

Let R be an o-minimal expansion of a real closed eld whose algebraic closure (in the sense of fields) is K. In earlier papers we investigated the notions of R-denable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quan- tiers after endowing it with all K-analytic subsets. Examples are compact complex manifolds and non-singular algebraic curves over K. We examine here basic properties of mild manifolds and prove that when a mild manifold M is strongly minimal and not locally modular then M is bi-holomorphic with a non-singular algebraic curve over K.

Mathematics Subject Classification: Primary 03C64, 03H05, 32P05; Secondary 03C98, 30F99, 32C25.

Keywords and phrases: O-minimality, Non-Archimedean complex analysis, Zariski geometries, Riemann existence theorem.

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