Model theory and algebraic geometry. An introduction to E. Hrushovski's proof of the geometric Mordell-Lang conjecture
The aim of the volume is to give a more or less self-contained exposition of E. Hrushovski's remarkable model-theoretic proof of the Mordell-Lang conjecture for function fields [\textit{E. Hrushovski}, J. Am. Math. Soc. 9, 667-690 (1996; Zbl 0864.03026)]. The conjecture can be roughly formulated as follows. Let \(k\) and \(K\) be algebraically closed fields, and \(k\) be a proper subfield of \(K\). Let \(A\) be an abelian variety and \(X\) its subvariety both defined over \(K\), and \(\Gamma\) be a subgroup of finite rank of \(A(K)\). Then either \(X\cap\Gamma\) is a finite union of cosets of subgroups of \(\Gamma\), or one can `descend' the whole situation to \(k\), by finding a homomorphism from \(A\) to an abelian variety \(S\) defined over \(k\) such that \(X\) is a translate of the preimage of a subvariety of \(S\) also defined over \(k\). For the case of characteristic zero the result had been proven by Yu. Manin in 1963; for the case of positive characteristic only partial results were known before Hrushovski's proof. Note that it is possible to deduce the result in characteristic zero from the corresponding result in positive characteristic. Hrushovski showed that this and some other questions of diophantine geometry can be naturally integrated in the abstract framework which has been developed in model theory. It turned out that the Mordell-Lang conjecture can be considered as an instance of a general model-theoretic trichotomy the idea of which had been discovered in the early eighties by B. Zilber. Note that more recently, by using a similar model-theoretic approach, Hrushovski found a new proof of the Manin-Mumford conjecture over a number field. The volume presents an exposition of Hrushovski's proof of the geometric Mordell-Lang conjecture as a series of coordinated papers written by experts in model theory (except one, written by M. Hindry, an algebraic geometer). To avoid technical difficulties, the presentation is restricted to the case of abelian varieties although Hrushovski's proof really works in the same way for semi-abelian varieties. Only the case of characteristic zero is presented exhaustively; for the case of positive characteristic only the details of the setting are given and the main differences between the two cases are explained. Some of the chapters are self-contained, and some are surveys. Here are the contents of the volume: \textit{E. Bouscaren}, ``Introduction to model theory'' (pp. 1-18); \textit{M. Ziegler}, ``Introduction to stability theory and Morley rank'' (pp. 19-44); \textit{D. Lascar}, ``Omega-stable groups'' (pp. 45-59); \textit{A. Pillay}, ``Model theory of algebraically closed fields'' (pp. 61-84); \textit{M. Hindry}, ``Introduction to abelian varieties and the Mordell-Lang conjecture'' (pp. 85-100); \textit{A. Pillay}, ``The model-theoretic content of Lang's conjecture'' (pp. 101-106); \textit{D. Marker}, ``Zariski geometries'' (pp. 107-128); \textit{C. Wood}, ``Differentially closed fields'' (pp. 129-141); \textit{F. Delon}, ``Separably closed fields'' (pp. 143-176); \textit{E. Bouscaren}, ``Proof of the Mordell-Lang conjecture for function fields'' (pp. 177-196); \textit{E. Hrushovski}, ``Proof of Manin's theorem by reduction to positive characteristic'' (pp. 197-205). In the whole, the collection is a very useful introduction to the relationship between model theory and algebraic geometry for mathematicians with only a basic knowledge of the two fields.
scientific article; zbMATH DE number 1206424