Mild manifolds and a non-standard Riemann existence theorem
type: Item
instance of: scholarly article
title: Mild manifolds and a non-standard Riemann existence theorem
zbMATH Open document ID: 1173.03035
author: S. Starchenko
author: Ya'acov Peterzil
published in: Selecta Mathematica. New Series
publication date: 2009-07-01T00:00:00Z
review text: 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.
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 32P05
zbMATH DE Number: 5574005
zbMATH Keywords: Riemann existence theorem
zbMATH Keywords: Zariski geometries
zbMATH Keywords: non-Archimedean complex analysis
zbMATH Keywords: o-minimality
MaRDI profile type: Publication
full work available at URL: s00029-008-0064-x
OpenAlex ID: W1980574208
DOI: 10.1007/S00029-008-0064-X
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description: scientific article; zbMATH DE number 5574005