The model theory of compact complex spaces

Abstract

The usual model-theoretic approach to complex algebraic geometry is to view complex algebraic varieties as living definably in the structure (C,+,×). Variousmodel-theoretic properties of algebraically closed fields (such as quantifier elimination and strong minimality) are then used to obtain geometric information about the varieties. This approach extends to other geometric contexts by considering expansions of algebraically closed fields to which the methods of stability or simplicity apply. For example, differential algebraic varieties live in differentially closed fields, and difference algebraic varieties in algebraically closed fields equipped with a generic automorphism. Another approach would be to consider the variety as a structure in its own right, equipped with the algebraic (respectively differential or difference algebraic) subsets of its cartesian powers. This point of view is compatible with the theory of Zariski-type structures, developed by Hrushovski and Zilber (see [21] and [41]). While the two approaches are equivalent (i.e., bi-interpretable) in the case of complex algebraic varieties, the latter point of view extends to certain fragments of complex analytic geometry in a manner that does not seem accessible by the former.

Cite this paper

@incollection{moosa2017model,
  title = {{The model theory of compact complex spaces}},
  author = {Moosa, Rahim},
  booktitle = {Cambridge University Press eBooks},
  pages = {317--349},
  year = {2017},
  doi = {10.1017/9781316755860.013}
}

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