Mild manifolds and a non-standard Riemann existence theorem
Let \({\mathcal R}\) be an o-minimal expansion of a real closed field \(R\) and \(K\) be the algebraic closure of \(R\), identified naturally with \(R^2\). In earlier papers, the authors investigated the notions of \(K\)-holomorphic maps and \(K\)-analytic manifolds definable in \({\mathcal R}\). Such a \(K\)-manifold is said to be mild if it eliminates quantifiers when endowed with all \(K\)-analytic subsets. Examples of mild \(K\)-manifolds are compact complex manifolds and non-singular algebraic curves over \(K\). The authors discuss here various properties of these manifolds. The main theorem ensures that a strongly minimal \(K\)-manifold is either locally modular or \(K\)-biholomorphic to a non-singular algebraic curve over \(K\). Note that the Riemann existence theorem does not hold in general for mild manifolds, so the authors cannot use it in their proof. However, they succeed in showing the second part of the Riemann existence theorem in \({\mathcal R}\) (just as an o-minimal expansion of a real closed field), which is enough for their purposes.
scientific article; zbMATH DE number 5574005