The First Order Theory of Universal Specializations Alf Onshuus and Boris Zilber

This paper concentrates on understanding the first order theory of universal specializations of Zariski structures. Models of the theory are pairs, a Zariski structure and an elementary extension with a map (specialization) from the extension to the structure that preserves positive quantifier free formulas. The reader will find that this context generalizes both the study of algebraically closed valued fields (see [2]) and sheds light on the theory of Zariski structures. It is also a natural setting for studying compact complex manifolds with a standard part map. We determine the first order theory of universal specializations and prove that it is model complete. Also, we prove that the ground Zariski structure in its core language is stably embedded in models of the theory, which has nice consequences for the theory of Zariski structures. The structure of the paper is as follows. In the rest of Section 1 we will give the basic definitions of Zariski structures. In Section 2 we will define specializations and prove that anything that can be defined in a Zariski structure using a universal specialization which projects onto it, can be defined within the original language (Corollary 2.11). Section 3 studies the consequences of Corollary 2.11 and is therefore a de-tour from the main goals of the paper, justified in the geometric implications that we can prove for the ground Zariski structure. Section 4 includes the main results of the paper: first order properties of universal specializations, including an axiomatization and a relative quantifier elimination. Finally, Appendix A will work a bit on realizations of types over universal specializations and their projections.