### 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 aleph_{0}-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.

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