Let R be an o-minimal expansion of a real closed field R, and K be the algebraic closure of R. In earlier papers we investigated the notions of R-definable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quantifers after endowing it with all it 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 it is biholomorphic to a non-singular algebraic curve over K.
Bibliographical noteFunding Information:
We would like to thank Elias Baro for pointing out a gap in one of the proofs. The second author was partially supported by the NSF.
- Non-Archimedean complex analysis
- Riemann existence theorem
- Zariski geometries
ASJC Scopus subject areas
- General Mathematics
- General Physics and Astronomy