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An introduction to motives. Pure motives, mixed motives, periods. (Une introduction aux motifs. Motifs purs, motifs mixtes, périodes.) (French) Zbl 1060.14001
Panoramas et Synthèses 17. Paris: Société Mathématique de France (ISBN 2-85629-164-3/pbk). xi, 261 p. (2004).
The concept of a motive in algebraic geometry is due to A. Grothendieck, who introduced it about 40 years ago as a tool to develop a systematic theory of arithmetic properties of algebraic varieties as embodied in their groups of classes of cycles. Grothendieck’s fundamental idea was to create motives as the building blocks of some universal, purely algebraic cohomology theory generalizing all other (known) cohomology theories. In other words, Grothendieck thought of a generalization of Galois theory to higher dimensions that would give rise to an abstract cohomology comprising all other cohomology theories as different concrete realizations. The reason behind his idea was that diverse cohomological phenomena, mysterious relationships, conjectures, and open problems involving integrals of algebraic functions (e.g., the famous Hodge conjecture) could be better understood by such a systematic universal approach. For a long time, the theory of motives remained merely conjectural, despite the spectacular developments it underwent during the last few decades. At its present stage, the theory of motives appears as a central topic of current research in algebraic and arithmetic geometry, and as a general framework that becomes more and more important and ubiquitous in many branches of pure mathematics. On the other hand, the theory of motives is still far from being part of the common mathematical knowledge, despite the avalanche of research articles on the subject and its tremendous progress in the past years. No doubt, the theory of motives is highly advanced, conceptionally rather complicated, technically involved, and a systematic, guiding textbook on, this topic is still lacking.
The book under review has been written to provide also non-specialists in the field with an introduction to the geometric foundations of the theory of motives, and with a panorama of some major developments that is has undergone in the recent 15 years. This is to help the mathematical community, as a whole, to keep track with the progress made in in this field, on the one hand, and to gain a systematic overview of its genesis, its present state of art, and its prospects on the other. Being a volume of the special French book series “Panoramas et Syntheses”, this book is written as a comprehensive, systematic and up-to-date survey on a developing topic in contemporary mathematics, without detailed proofs, but with a plentiful supply of hints at the sources and the current literature for detailed reading. Also, the author explains and illustrates the fundamental concepts and results of the geometric theory of motives, together with their interrelations and concrete applications.
The text consists of three parts, each of which is divided into numerous sections and subsections.
Part I is devoted to pure motives and contains the first thirteen sections. The wealth of information brought about in this part includes digests of such topics as tensor categories, Tannakian categories, algebraic cycles, Weil cohomologies, pure motives à la Grothendieck, numerical motives, weights, the standard conjectures motivic Galois groups, the conjectures of Hodge, Tate, and Ogus and their generalizations, filtrations of Chow rings, the conjectures of Bloch-Beilinson-Murre, Voevodsky’s nilpotency conjecture, the category of motives, Kimura-O’Sullivan categories, Chow motives, and motivic zeta functions.
Part II comprises the following nine chapters and focuses on mixed motives. Again, a plenty of topics is touched upon in this context, among which are the following central ones: weight filtrations, mixed motives, motivic cohomology, the formalism of multivalued morphisms, the Suslin-Voevodsky construction, Suslin homology, Voevodsky’s mixed motives, the categories \(\text{DM}^{\text{eff}}_{\text{gm}}\) and \(\text{DM}_{\text{gm}}(k)\), comparisons with Chow groups and higher algebraic K-theory, Mayer-Vietoris triangles and Gysin triangles, complexes of motivic sheaves, the Nisnevich topology and transfers, 1-motives and mixed Tate motives, Kummer motives, the Hard Lefschetz Theorem in motivic cohomology, the conjectures of Bloch-Beilinson-Murre revisited, the category of Nori’s mixed motives, mixed realizations and regulators, values of \(L\)-functions, and much more.
Part III, the last part of the book, turns to the study of periods of motives. This is the shortest part (Chapters 23 to 25), and the author puts here special emphasis on illustrations and examples of the abstract theory of motives. Fields of periods, Grothendieck’s conjecture on periods of integrals, its extension to mixed motives, the idea of a transcendental Galois theory are discussed in the beginning, followed by a more detailed account of values of the gamma function at rational points (Chapter 24) and, in the sequel, by an explanation of the relation between motives and polyzeta numbers in the concluding Chapter 25.
The extremely rich and nearly complete bibliography, which the author ponders over throughout the concise but comprehensive survey, refers to more than 180 titles. It is a huge amount of material that the reader gets acquainted with, but the author has arranged it in a highly systematic, enlightening and and masterly manner. Numerous additional remarks and cross-references help the reader digest that abundance of extremely advanced material, and even some proofs are indicated whenever they are reasonably short.
At any rate, the book under review is the most systematic and comprehensive survey of the theory of motives so far, and a perfect guide into deeper studies as well. Likewise, it is a profound source and reference book for active researchers in the field.

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
11J91 Transcendence theory of other special functions
14F42 Motivic cohomology; motivic homotopy theory
14C25 Algebraic cycles