##
**Geometric stability theory.**
*(English)*
Zbl 0871.03023

Oxford Logic Guides. 32. Oxford: Clarendon Press. x, 361 p. (1996).

Model-theoretic stability theory started in the early sixties with Morley’s seminal paper in which he proved that a complete countable first-order theory is categorical in every uncountable power provided it is categorical in some uncountable power. The technique he developed in the proof turned out to be important in its own right. Starting from it, Shelah initiated a program of classification of theories asking whether or not one can prove a ‘structure’ theorem for the class of models of the theory. One of the most important notions in Shelah’s theory is stability. A first-order theory is said to be stable if no model of it contains an infinite set of tuples on which some formula defines a linear ordering. Shelah showed that an unstable theory \(T\) has \(2^\lambda\) models of cardinality \(\lambda\) for all \(\lambda >|T|\), and so, from the point of view of his program, no structure theorem can be proved for unstable theories. For stable theories, he developed a sophisticated machinery to pursue the program. By using this machinery, in 1982 he found a solution of the structure/nonstructure problem for complete countable first-order theories. Shelah’s theory is published in his book [S. Shelah, Classification theory and the number of non-isomorphic models, 2nd edition (1990; Zbl 0713.03013)]. There are several books where various parts of the theory are presented in a readable and coherent form [A. Pillay, An introduction to stability theory (1983; Zbl 0526.03014); B. Poizat, Cours de théorie des modèles (1985; Zbl 0583.03001); D. Lascar, Stabilité en théorie des modèles (1986; Zbl 0655.03021); J. T. Baldwin, Fundamentals of stability theory (1988; Zbl 0685.03024); S. Buechler, Essential stability theory (1996; Zbl 0864.03025)].

In the seventies and the early eighties Zilber developed a series of results and conjectures about uncountably categorical theories, the key conjecture being the possibility of classifying such theories up to ‘bi-interpretability’. Baldwin and Lachlan proved that models of any such a theory contain certain ‘irreducible’ definable sets, the so-called strongly minimal sets; as Marsh earlier showed, any strongly minimal set forms a pregeometry with respect to the algebraic closure operator. Zilber conjectured that it is possible to classify the pregeometries arising in this way and then go on to classify uncontably categorical structures; in a series of beautiful and profound papers he showed how to realize this idea for totally categorical theories. His results are summarized in his book [B. Zilber, Uncountably categorical theories (1993; Zbl 0785.03019)], which is a translation of his D. Sc. Thesis (Kemerovo University, 1986). Many of the ideas developed by Zilber under the assumption of uncountable categoricity turned out to be useful in the stable framework. Zilber’s conjectures and work represented the birth of geometric stability theory that became one of the mainstreams in model theory in the last two decades. This theory deals with the interaction between ‘global’ properties of stable theories (such as the global geometric behaviour of the forking independence relation) and ‘local’ properties (such as the behaviour of the independence relation on the sets of realizations of some special types). The book under review, written by one of the main contributors to the field, is the first textbook in the literature where main results of geometric stability theory are presented in a coherent form.

Chapter 1, Stability theory, is a survey, with selective proofs, of stability theoretic results and notions required for the rest of the book (forking, canonical bases, ranks, orthogonality, stable group theory). In Chapter 2, The classical finite rank theory, the author develops the ‘classical’ part of geometric stability theory, consisting essentially of results of Zilber, Cherlin-Harrington-Lachlan, and Buechler. Main results here are local modularity of \(\omega\)-categorical strongly minimal sets, the theorem that a weakly minimal stationary type is locally modular or has Morley rank 1, ‘coordinatization’ in theories of finite \(U\)-rank, and non-finite axiomatizability of totally categorical complete theories. Chapter 3, Quasifinite axiomatizability, deals with the theorem due to Ahlbrandt-Ziegler (in a special case) and Hrushovski that \(\omega\)-categorical \(\omega\)-stable theories are finitely axiomatizable modulo a finite number of ‘schemas of infinity’. In Chapter 4, 1-based theories and groups, the author considers 1-based theories – stable theories with good global behaviour of the independence relation; for a superstable theory of finite \(U\)-rank, 1-basedness is equivalent to local modularity of all minimal types. It is shown that under various additional assumptions (triviality, NDOP, few types) 1-basedness implies that the theory is ‘superstable-like’. The results on groups definable in 1-based theories due to Pillay-Hrushovski are presented; these groups turned out to be essentially abelian groups equipped with predicates for subgroups. In Chapter 5, Groups and geometries, it is shown that one can recognize the presence of \(\infty\)-definable group in a stable theory from a certain geometric configuration of points. By using this, it is proved that the geometry associated with a minimal set is the affine or projective geometry over a division ring provided it is locally modular and nontrivial. These results are due to Hrushovski; the approach comes from Zilber, who worked in \(\omega\)-categorical strongly minimal context. In Chapter 6, Unidimensional theories, the author gives applications of geometric stability theory to classification-theory-type problems: Laskowski’s result concerning the models of an uncountable theory \(T\) which is categorical in a power \(>|T|\), and a proof of the Vaught Conjecture for weakly minimal theories due to Buechler and Newelski. Chapter 7, Regular types, contains Hrushovski’s analysis of geometries attached to regular types, including a characterization of them, in the locally modular case, as degenerate, or affine, or projective geometries over division rings. In Chapter 8, Superstable theories, the author presents some technique which helps generalizing results from the finite rank context to the general superstable context; main results here are due to Hrushovski-Shelah and Chowdhury-Pillay. (Some of this machinery turned out to be useful in Hrushovski’s precise computation of the possible spectrum functions for countable superstable shallow theories with NDOP and NOTOP, which completes Shelah’s results of the possible spectrum functions for countable complete theories; a full presentation of the result is to appear in a forthcoming paper by Hart-Hrushovski-Laskowski.) One of the results of Chapter 8 is Hrushovski’s theorem on superstability of any unidimensional theory.

The book contains careful historical notes and references. Among the important developments that have not been included to the book are the theory surrounding Hrushovski’s constructions of new strongly minimal sets, the theory of Zariski geometries due to Hrushovski-Zilber, applications of geometric model theory to diophantine geometry due to Hrushovski, and recent results on the so-called simple theories due to Kim-Pillay, which show that many of the ideas of stability theory go beyond the stable context.

The book under review is a much needed and highly recommended textbook in a beautiful, deep, and actively developing field. However, it is not an easy reading for beginners: the reader is assumed to have some experience in model theory, the material is quite dense, and requires substantial work from the reader.

In the seventies and the early eighties Zilber developed a series of results and conjectures about uncountably categorical theories, the key conjecture being the possibility of classifying such theories up to ‘bi-interpretability’. Baldwin and Lachlan proved that models of any such a theory contain certain ‘irreducible’ definable sets, the so-called strongly minimal sets; as Marsh earlier showed, any strongly minimal set forms a pregeometry with respect to the algebraic closure operator. Zilber conjectured that it is possible to classify the pregeometries arising in this way and then go on to classify uncontably categorical structures; in a series of beautiful and profound papers he showed how to realize this idea for totally categorical theories. His results are summarized in his book [B. Zilber, Uncountably categorical theories (1993; Zbl 0785.03019)], which is a translation of his D. Sc. Thesis (Kemerovo University, 1986). Many of the ideas developed by Zilber under the assumption of uncountable categoricity turned out to be useful in the stable framework. Zilber’s conjectures and work represented the birth of geometric stability theory that became one of the mainstreams in model theory in the last two decades. This theory deals with the interaction between ‘global’ properties of stable theories (such as the global geometric behaviour of the forking independence relation) and ‘local’ properties (such as the behaviour of the independence relation on the sets of realizations of some special types). The book under review, written by one of the main contributors to the field, is the first textbook in the literature where main results of geometric stability theory are presented in a coherent form.

Chapter 1, Stability theory, is a survey, with selective proofs, of stability theoretic results and notions required for the rest of the book (forking, canonical bases, ranks, orthogonality, stable group theory). In Chapter 2, The classical finite rank theory, the author develops the ‘classical’ part of geometric stability theory, consisting essentially of results of Zilber, Cherlin-Harrington-Lachlan, and Buechler. Main results here are local modularity of \(\omega\)-categorical strongly minimal sets, the theorem that a weakly minimal stationary type is locally modular or has Morley rank 1, ‘coordinatization’ in theories of finite \(U\)-rank, and non-finite axiomatizability of totally categorical complete theories. Chapter 3, Quasifinite axiomatizability, deals with the theorem due to Ahlbrandt-Ziegler (in a special case) and Hrushovski that \(\omega\)-categorical \(\omega\)-stable theories are finitely axiomatizable modulo a finite number of ‘schemas of infinity’. In Chapter 4, 1-based theories and groups, the author considers 1-based theories – stable theories with good global behaviour of the independence relation; for a superstable theory of finite \(U\)-rank, 1-basedness is equivalent to local modularity of all minimal types. It is shown that under various additional assumptions (triviality, NDOP, few types) 1-basedness implies that the theory is ‘superstable-like’. The results on groups definable in 1-based theories due to Pillay-Hrushovski are presented; these groups turned out to be essentially abelian groups equipped with predicates for subgroups. In Chapter 5, Groups and geometries, it is shown that one can recognize the presence of \(\infty\)-definable group in a stable theory from a certain geometric configuration of points. By using this, it is proved that the geometry associated with a minimal set is the affine or projective geometry over a division ring provided it is locally modular and nontrivial. These results are due to Hrushovski; the approach comes from Zilber, who worked in \(\omega\)-categorical strongly minimal context. In Chapter 6, Unidimensional theories, the author gives applications of geometric stability theory to classification-theory-type problems: Laskowski’s result concerning the models of an uncountable theory \(T\) which is categorical in a power \(>|T|\), and a proof of the Vaught Conjecture for weakly minimal theories due to Buechler and Newelski. Chapter 7, Regular types, contains Hrushovski’s analysis of geometries attached to regular types, including a characterization of them, in the locally modular case, as degenerate, or affine, or projective geometries over division rings. In Chapter 8, Superstable theories, the author presents some technique which helps generalizing results from the finite rank context to the general superstable context; main results here are due to Hrushovski-Shelah and Chowdhury-Pillay. (Some of this machinery turned out to be useful in Hrushovski’s precise computation of the possible spectrum functions for countable superstable shallow theories with NDOP and NOTOP, which completes Shelah’s results of the possible spectrum functions for countable complete theories; a full presentation of the result is to appear in a forthcoming paper by Hart-Hrushovski-Laskowski.) One of the results of Chapter 8 is Hrushovski’s theorem on superstability of any unidimensional theory.

The book contains careful historical notes and references. Among the important developments that have not been included to the book are the theory surrounding Hrushovski’s constructions of new strongly minimal sets, the theory of Zariski geometries due to Hrushovski-Zilber, applications of geometric model theory to diophantine geometry due to Hrushovski, and recent results on the so-called simple theories due to Kim-Pillay, which show that many of the ideas of stability theory go beyond the stable context.

The book under review is a much needed and highly recommended textbook in a beautiful, deep, and actively developing field. However, it is not an easy reading for beginners: the reader is assumed to have some experience in model theory, the material is quite dense, and requires substantial work from the reader.

Reviewer: Oleg V. Belegradek (Kemerovo)

### MSC:

03C45 | Classification theory, stability, and related concepts in model theory |

03-02 | Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations |