MILD MANIFOLDS AND A NON-STANDARD RIEMANN EXISTENCE THEOREM

Abstract. 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 no-tions of R-definable K-holomorphic maps, K-analytic manifolds and their K-analytic subsets. We call such a K-manifold mild if it eliminates quan-tifiers 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. 1. introduction For a complex-analytic manifold M, letM be the first-order structure whose underlying set is M and basic relations are all complex-analytic subsets of Carte-sian powers of M. The following result is due to Zilber (see [17]). Theorem 1.0.1. If M is a compact complex-analytic manifold, then M has elim-ination of quantifiers and has finite Morley rank. If M is strongly minimal then it is a Zariski geometry (see [8] for a definition of Zariski geometries). Let R be a fixed o-minimal expansion of a real closed field R, and let K = R( √−1) be its algebraic closure identified naturally with R2. In a series of papers [12–14] we investigated notions K-holomorphic maps and K-analytic manifolds definable in the structure R. For a K-analytic manifold M let A(M) be the first order structure whose underlying set is M and basic relations are all K-analytic subsets of Cartesian powers of M (see Section 2.2 for definitions of K-analytic manifolds and K-analytic subsets). In paper [14] we showed that a version of the above Zilber’s theorem holds in this general setting