Analytic Zariski Structures and Non-elementary Categoricity

We study analytic Zariski structures from the point of view of non-elementary model theory. We show how to associate an abstract elementary class with a one-dimensional analytic Zariski structure and prove that the class is stable, quasi-minimal and homogeneous over models. We also demonstrate how Hrushovski’s predimension arises in this general context as a natural geometric notion and use it as one of our main tools. The notion of an analytic Zariski structure was introduced in [1] by the author and N.Peatfield in a form slightly different from the one presented here and then in [2], Ch.6 in the current form. Analytic Zariski generalises the previously known notion of a Zariski structure (see [2] for one-dimensional case and [3], [4] for the general definition) mainly by dropping the requirement of Noetherianity and weakening the assumptions on the projections of closed sets. We remark that in the broad setting, it is appropriate to consider the notion of a Zariski structure as belonging to positive model theory in the sense of [5]. In [1] we assumed that the Zariski structure is compact (or compactifiable), here we drop this assumption, which may be too restrictive in applications. The class of analytic Zariski structures is much broader and geometrically richer than the class of Noetherian Zariski structures. The main examples come from two sources: (i) structures which are constructed in terms of complex analytic functions and relations; (ii) “new stable structures” introduced by Hrushovski’s construction; in many cases these objects exhibit properties similar to those of class (i). However, although there are concrete examples for both (i) and (ii), in many cases we lack the technology to prove that the structure is analytic Zariski. In particular, despite some attempts the conjecture that Cexp is analytic Zariski, assuming it satisfies axioms of pseudo-exponentiation (see [17]), is still open. The aim of this paper is to carry out a model-theoretic analysis of M in the appropriate language. Recall that if M is a Noetherian Zariski structure the relevant key model-theoretic result states that its first-order theory allows elimination of quantifiers and is ω-stable of finite Morley rank. In particular, it is strongly minimal (and so uncountably categorical) if dimM = 1 and M is irreducible. For analytic Zariski 1-dimensional M we carry out a model theoretic study it in the spirit of the theory of abstract elementary classes. We start by introducing a suitable countable fragment of the family of basic Zariski relations and a correspondent substructure of constants over which all the further analysis is carried Date: 12 January, 2016.