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Complex abelian varieties. (English) Zbl 0779.14012
Grundlehren der Mathematischen Wissenschaften. 302. Berlin: Springer- Verlag. viii, 435 p. (1992).
The history of the theory of abelian varieties is one of the most fascinating examples of the tremendous development of mathematics, as a whole, during the last two centuries. Its roots can be traced back to the beginning of the 19th century, when N. H. Abel and C. G. J. Jacobi discovered an approach to tackle the old problem of integrating multivalued functions on the complex plane. In geometrical terms, they worked with what is now called the Riemann surface of a multivalued holomorphic function, constructed (implicitly) its Jacobian torus, the associated (nowadays so-called) Abel-Jacobi mapping from the surface into its Jacobian, and studied multivalued functions and their integrals as single-valued functions (and their integrals) on those Jacobi tori. Later on, B. Riemann pushed the theory further, namely by using theta functions as a basic tool for investigating abelian functions, and a the end of the 19-th century mathematicians such as K. Weierstrass, H. Poincaré, E. Picard, G. Frobenius, F. Prym, F. Schottky, A. Krazer, W. Wirtinger and others made the whole subject of abelian functions a culmination point in complex function theory. At the beginning of the 20-th century, M. de Franchis, H. Baker, S. Lefschetz, C. Rosati, G. Scorza and others invented the geometric viewpoint in the study of abelian functions and theta functions, in that they investigated the images of $$\mathbb{C}^ g$$ in projective space, under mappings defined by linearly independent theta functions. These geometric objects, as projective varieties, originated the study of particular nonsingular complex group varieties, which we call today complex abelian varieties.
Although abelian varieties are of great significance in several branches of mathematics, and despite the fact that the subject has undergone an enormous development during the past 50 years, there are just a few textbooks providing both a profound introduction and a reasonably comprehensive, updated presentation of the current stage of the subject. This has been regrettable for a long time, all the more as abelian varieties have become an indispensible tool in arithmetic geometry and algebraic number theory, especially during the past 10 years, also a crucial ingredient for analysing integrable Hamiltonian systems and nonlinear evolution equations in mathematical physics (Schrödinger equations, Korteweg-de Vries equation, Kadomtsev-Petviashvili equation, Zakharov-Shabat equations, Lax equations, Novikov-Veselov equation, etc.) and, as for the own sake of algebraic geometry, abelian varieties occur as associated objects to smooth projective varieties, e.g., as Picard varieties, Albanese varieties, Griffiths’s intermediate Jacobians and Prym varieties. As such, they are particularly important for investigating rationality and transcendence questions on varieties and (birational) classification problems.
For more than 20 years, D. Mumford’s classic “Abelian varieties” [Tata Inst. Fund. Res. Stud. Math. 5 (London 1970; Zbl 0223.14022)] was the only modern textbook on the subject, but it mainly focused on general abelian varieties (over arbitrary groundfields). Then came D. Mumford’s beautiful series “Tata lectures on theta. I, II, III” [Prog. Math. 28(1983), 43(1984), 97 (1991); respectively Zbl 0509.14049, Zbl 0549.14014, Zbl 0744.14033] stressing theta functions and their ubiquity in today’s mathematics (more than the geometry of complex abelian varieties) and, recently, G. Kempf’s consise (but very profuse) booklet “Complex abelian varieties and theta functions” (1991; Zbl 0752.14040). Of course, there were some other textbooks dealing with complex abelian varieties, for example F. Conforto’s still lovely one “Abelsche Funktionen und algebraische Geometrie” (1956; Zbl 0098.132; reprint (1983; Zbl 0074.366), S. Lang’s “Abelian varieties” (1958; Zbl 0516.14031) and “Introduction to algebraic and abelian functions” (1972; Zbl 0255.14001; 2nd edition 1982); Zbl 0513.14024) as well as H. P. F. Swinnerton-Dyer’s “Analytic theory of abelian varieties” (1974; Zbl 0299.14021), but all of them presented mainly the classical material and were not comprehensive whatsoever. In addition, there was no textbook, until now, which provided a full account on Jacobian varieties and Prym varieties within the general, modern theory of complex abelian varieties.
The present book under review aims to remedy this state of things, and to present both a modern, comprehensive textbook and a firm reference book for current research in the field. It attempts to cover the main methods and results of the theory of complex abelian varieties, from classical period to very recent, in the contemporary language of algebraic and complex-analytic geometry, and to focus, in particular and for the first time, on those aspects that are scattered in the recent research literature and have never been treated systematically in a monography, much less in a textbook. Certainly, to write such a systematic monography on a longstanding, vividly developing and beautifully intricate topic is a very ambitious and admirable undertaking. The authors managed that perfectly, contributing a great deal of their own recent research and results, and they succeeded in producing a nearly self-contained book that leaves nothing to be desired concerning comprehensiveness, systematic representation, clarity, rigor, modernity, and inspiring illumination.
The content consists of twelve chapters and two appendices. The first three chapters cover the classical material on complex tori, their cohomology and Hodge theory, their line bundles and Poincaré bundles, their theta functions, and the cohomology of line bundles on tori, including the Riemann-Roch theorem and the classical vanishing theorems. – In chapter 4 the authors start the discussion of complex abelian varieties, i.e., of complex tori admitting a positive definite line bundle. This encompasses the discussion of polarizations, the Riemann relations, the decomposition theorem, the main results about the Gauss map of a divisor, their applications to projective embeddings (Lefschetz’s theorem) and Kummer varieties, and – absolutely new in a book – a detailed study of symmetric line bundles and divisors. The chapter concludes with the basic facts on maps into abelian varieties and the various equivalence relations for algebraic cycles. – Chapter 5 deals with endomorphisms of abelian varieties and includes the treatment of the Rosati involution, the Néron-Severi group, Poincaré’s complete reducibility theorem, and the endomorphism algebra of a simple abelian variety.
Chapter 6 provides a detailed, systematic account on D. Mumford’s theory of theta groups and Heisenberg groups, together with their representations. This chapter is essentially self-contained, but heavily used in the sequel. Namely, the following chapters deal with the central topics of the book: the projective embeddings of an abstract abelian variety, the equations defining an embedded abelian variety, and geometric properties of abelian varieties, and the main tool for the proofs is the theta group of a line bundle, just as the concept of the characteristic of a nondegenerate line bundle. These parts of the book contain old and new results on embedded abelian varieties and the equations defining them, whereas the general theory (over arbitrary fields) was essentially developed by D. Mumford [cf. “On the equations defining abelian varieties. I, II, III”, Invent. Math. 1, 287- 354 (1966); 3, 75-135 and 215-244 (1967; Zbl 0219.14024)]. – More precisely, chapter 7 is devoted to the equations defining abelian varieties in projective space and contains the multiplication formula, a discussion on projective normality, Riemann’s theta relations and the cubic theta relations. This topic is taken up again in chapter 10, where abelian surfaces – the first interesting case after the classical work on elliptic curves – are the objects of study. Here the authors provide the first textbook presentation of some of the most recent results, such as the geometry of the Kummer surface [cf. K. Hulek and H. Lange, J. Reine Angew. Math. 363, 201-216 (1985; Zbl 0593.14027)] and Reider’s theorem [cf. I. Reider, Ann. Math., II. Ser. 127, No. 2, 309-316 (1988; Zbl 0663.14010)].
In between, chapters 8 and 9 discuss the moduli problem for complex abelian varieties (with level structures) from the analytic viewpoint (i.e., via Siegel spaces and Siegel modular functions), including theta relations, and (in chapter 9) moduli spaces for abelian varieties with distinguished endomorphism structure. The latter topic includes a thorough treatment of real multiplication, complex multiplication, quaternion multiplication, families of abelian varieties admitting such kinds of multiplication, and a brief introduction to Shimura varieties. All this gives a systematic account on Shimura’s theory [cf. Y. Matsushima and G. Shimura, Ann. Math., II. Ser. 78, 417-449 (1963; Zbl 0141.387)], embedded into the general framework.
Chapter 11 provides the basic theory of Jacobian varieties. The authors focus on results and proofs that illustrate the general theory developed before and, in addition and rewardingly, on results which have not been taken up yet in other books. Thus the main topics are: a version of Fay’s trisecant formula for Kummer varieties of Jacobians, Z. Ran’s improvement of Matsusaka’s criterion for characterizing Jacobians geometrically, and the Riemann-Kempf singularity theorem for theta divisors.
The concluding chapter 12 deals with another important example of abelian varieties: the Prym varieties associated with double coverings of smooth complex curves. It provides, for the first time in a textbook, an introduction to the theory of Prym varieties, introduced (implicitly) by W. Wirtinger in 1895, after F. Prym’s discovery of special differentials associated with double coverings, and algebraically generalized by D. Mumford [in Contribut. to Analysis, Collect. Pap. dedic. L. Bers, 325-350 (1974; Zbl 0299.14018)]. One can say that this chapter gives a rather complete account of the present state of the theory of Prym varieties, encompassing the structure of the Abel-Prym map, the geometry of the Prym theta divisor, Recillas’s theorem, Donagi’s tetragonal construction, and Kanev’s criterion for defining Prym-Tyurin varieties. All these results are fairly recent, and firstly included in a textbook.
Appendix A compiles the basic facts about algebraic varieties and complex analytic spaces, as they are used in the text, and appendix B explains the method of defining line bundles over complex manifolds by factors of automorphy with respect to the fundamental group.
Not covered in this book are some important topics like the Schottky problem of characterizing Jacobians (or Prym varieties), intermediate Jacobians, compactifications of moduli spaces, families and degenerations of abelian varieties, etc. The authors are completely right when they say that each of these extremely rich and intricate subjects would require an extra volume. Anyway, the reader is amply requited with the present outstanding text, and he will be well-equipped for further studies of abelian varieties and their applications in various branches. Each chapter goes with its own motivating introduction, and ends with a number of exercises of varying degrees of difficulty. Many of them intend to complete the material in the text by further standard results, others contain more recent and advanced results from the current research literature; they are an invitation to further reading.
As for the prerequisites, the reader should have some basic knowledge in algebraic geometry and complex analysis, including cohomology and sheaf theory, but all in all the text is fairly self-contained. – Altogether, this book fills a gap in the literature on complex abelian varieties and their applications. It is a great reference and textbook, detailed, very up-to-date, thorough, clearly written and perfectly arranged. It is just a standard book, recommended to everybody interested or working in this fascinating field.

##### MSC:
 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14K25 Theta functions and abelian varieties 14K30 Picard schemes, higher Jacobians 14K10 Algebraic moduli of abelian varieties, classification 14H40 Jacobians, Prym varieties