Markus Junker "Géometries de Zariski et Groupes de Zariski", PhD thesis, Paris, July 1996.


Abstract

Zariski geometries were introduced by E. Hrushovski and B. Zil'ber as abstract models of algebraic curves to get a class of structures where the Zil'ber trichotomy holds. In my thesis, I define a higher dimensional generalization of the notion of Zariski geometry and I start the systematic study of these structures.

A Zariski geometry is given by a family of Noetherian topologies such that the topological dimension shares several properties of the concept of dimension in algebraic geometry (chapter 1), and such that the essential properties are preserved under elementary extensions. It follows that Zariski geometries are special aleph0-stable structures where topological and model theoretical properties are closely linked (chapter 2).

Besides trivial and linear structures, algebraic varieties over algebraically closed fields form the most important class of examples (chapter 4).

A structural description of Zariski geometries requires a deep knowledge of the interpretable structures. Chapter 3 of this thesis shows that the topological behaviour of imaginary sets is quite complex. This analysis leads to the definition of a variety over a Zariski geometry and to the appropriate notion of morphisms. Special important types of varieties are smooth and complete varieties. They are defined and studied separately.

As an application of the techniques developed so far, I prove a structure theorem for Zariski groups (defined over Zariski geometries as algebraic groups over fields), namely that there are no bad smooth Zariski groups. Combinig this result with a theorem of Zil'ber, one concludes that every simple smooth Zariski group interprets an algebraically closed field. This proves one half of Cherlin's conjecture for smooth Zariski groups and almost provides an abstract characterization of simple algebraic groups over algebraically closed fields.


[back]