In the last two decades or so a bridge has been built between algebraic geometry and the model theory of stable algebraic structures. An important part of this bridge is an axiomatization of the geometry underlying algebraic varieties and complex-analytic manifolds, the so-called Zariski geometries of E. Hrushovski and B. Zilber (see for example “Zariski Geometries”, J. Amer. Math. Soc. 9 (1996), 1–56 and Bull. Amer. Math. Soc. 28 (1993), 315–323). This interaction has brought several important applications of model theory to Diophantine geometry. One such application is Hrushovski’s proof of the Mordell-Lang conjecture for function fields (J. Amer. Math. Soc. 9 (1996), 667–690).
Perhaps the starting point of these developments is Tarski’s theorem on elimination of quantifiers for the theory of algebraically closed fields: It is known that this is equivalent to Chevalley’s theorem in algebraic geometry (the image of a constructible set is constructible). But the turning point was Morley’s theorem, the beginning of stability theory. Morley’s theorem characterizes the isomorphism type of a structure by its model-theory description and its cardinality. Stability gives rise to a hierarchy of theories, with theories of finite Morley rank and categorical for uncountable cardinals at the top of this hierarchy. Classifying such theories is a central goal of model theory. It was when working towards this classification that certain algebro-geometric ideas have proved to be essential, since purely logical conditions were proved to be insufficient. This is when the notion of a Zariski structure came along.
Roughly speaking, a Zariski geometry specifies a family of relations on the structure M, requiring that for each nonnegative integer n, the subsets of the Cartesian product Mn satisfy the axioms for a Noetherian topology; a notion of dimension for M is also required. Examples of Zariski geometries are, of course, the Zariski topology on an algebraic variety over an algebraically closed field, where the dimension is the classical Krull dimension. Another class of examples is given by compact complex manifolds with the topology given by the analytic subsets. One further example corresponds to the rigid varieties over non-archimedean fields. For a finite group G acting on an algebraic variety M, the set of orbits M/G with the natural topology is a Zariski structure: an orbifold. Notice that, in general, the orbifold M/G is not an algebraic variety.
A central, and still open, problem is the classification of Zariski geometries. This has already been done in the one-dimensional case where it is shown that one-dimensional Zariski geometries are essentially algebraic curves over an algebraic closed field. The book under review devotes Chapters 3 and 4 to proving this theorem. Some facts from model theory are recalled in Chapter 1, and Chapter 2 sets the basic notions of topological structures (compact Noetherian topologies, irreducibility, constructible sets) into the formal language of mathematical logic. Chapter 5 deals with some examples of non classical Zariski structures, e.g., Zariski structures that are not interpretable as algebraic varieties over algebraically closed fields. Chapter 6 is devoted to some generalized Zariski structures that are obtained by weakening or dropping some conditions, for instance by dropping the Noetherian condition.
The book has two appendices. The first one collects some general facts on formal languages and interpretations, a few results from model theory (e.g., the compactness theorem, and the Löwenheim-Skolem theorem), model completeness and categoricity, illustrating these concepts with examples that are used throughout the book. For the important example of the theory of algebraically closed fields on p. 174, the notions of trascendence basis and trascendence degree should be amended to take into account that these are relative notions, and so Steinitz’s theorem on the same page should also be corrected. The second appendix collects some useful results on geometric stability theory.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org
1. Introduction; 2. Topological structures; 3. Noetherian Zariski structures; 4. Classification results; 5. Non-classical Zariski geometries; 6. Analytic Zariski geometries; A. Basic model theory; B. Geometric stability theory; References; Index.