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Complex abelian varieties. 2nd augmented ed. (English) Zbl 1056.14063
Grundlehren der Mathematischen Wissenschaften 302. Berlin: Springer (ISBN 3-540-20488-1/hbk). xii, 635 p. (2004).
The first edition of this uniquely comprehensive monograph [“Complex abelian varieties”, Berlin, Springer-Verlag (1992; Zbl 0779.14012)] on the extensive theory of complex abelian varieties appeared in 1992. Back then, it has been reviewed and praised at length, and its nearly encyclopedic character has been thoroughly described. In fact, this combination of detailed textbook, up-to-date research monograph and systematic reference book on complex abelian varieties has immediately made it the unrivalled standard text in its field, and this exceptional position of the book in the current literature has remained undisputed until today.
However, in the course of the last fifteen years, the theory of abelian varieties has undergone significant further developments, with remarkable progress in various directions. The book under review is the second, essentially augmented edition of the original standard text, which now also reflects some of the very recent developments. The authors have added to this second edition five new chapters and four supplementary appendices, thereby increasing the volume of the book by 200 additional pages, or at a rate of 46 percent, respectively. And, of course, the bibliography has been accordingly up-dated and enhanced.
The original text consisted of twelve chapters and two appendices, all of which have been left entirely intact, apart from the correction of a few errors pointed out to the authors in the meantime. As to the five new chapters, now numbered as Chapters 13 to 17, their contents are arranged as follows:
Chapter 13 is entitled “Automorphisms” and deals with old and new results on automorphism groups of polarized abelian varieties. This include various fixed-point formulae for automorphisms, abelian varieties of CM-type, abelian surfaces with finite automorphism groups, Poincaré’s reducibility theorem for abelian varieties with automorphisms, and the isogeny decomposition of an abelian variety acted on by a group of automorphisms. This chapter also leads into the arithmetic theory of abelian varieties (of CM-type), using (without proof this time) classical results of G. Shimura and Y. Taniyama [Publications of the Mathematical Society of Japan, 6. Tokyo: The Mathematical Society of Japan. XI. (1961; Zbl 0112.03502)].
Chapter 14 discusses vector bundles on abelian varieties. The authors give an introduction to this topic, which has been tremendously propelled by S. Mukai’s invention of what is now known as the Fourier-Mukai transform [Nagoya Math. J. 81, 153–175 (1981; Zbl 0417.14036)]. Of course, a full account of this extensive categorical framework would be beyond the scope of the present all-round text, and therefore the authors frequently refer to the recent special monograph on this advanced topic by A. Polishchuk [“Abelian varieties, theta functions and the Fourier transform”, Cambridge Tracts in Mathematics 153, Cambridge University Press, Cambridge (2003; Zbl 1018.14016)] as for further details and results. However, stressing the didactical point of view, and providing purposeful ad-hoc constructions, they masterly describe the geometry of the Poincaré bundle, Picard sheaves, vector bundles on abelian surfaces, and various concrete applications. Chapter 15 is devoted to further results on line bundles and the theta divisor on an abelian variety. Basically, this chapter completes the material exhibited in Chapter 7 and Chapter 10, thereby focusing on three main subjects: syzygies of line bundles, Seshadri constants of ample line bundles, and the singularities of the theta divisor, culminating in a proof of the recent theorem by L. Ein and R. Lazarsfeld [J. Am. Math. Soc. 10, No. 1, 243–258 (1997; Zbl 0901.14028)].
Chapter 16 turns to the theory of algebraic cycles on abelian varieties and its recent achievements. After introducing Chow groups, the authors explain the Fourier-Mukai transform on both the Chow ring and the cohomology ring of an abelian variety, the Künneth decomposition for Chow rings, and the so-called Bloch filtration of the group of zero-dimensional cycles.
Finally, Chapter 17 continues the discussion of algebraic cycles with a view toward the famous Hodge conjecture for abelian varieties. This includes the necessary background material on Hodge structures and complex structures, the description of the Siegel upper-half space in terms of symplectic complex structures, and the analysis of the Hodge group of an abelian variety as well as A. Mattuck’s structure theorem on the Hodge ring of a general polarized abelian variety. This chapter (and the book) concludes with an exemplary proof of the Hodge \((p,p)\)-conjecture for a general Jacobian variety, which appears as a special case of a more general result recently proved by I. Biswas and M. S. Narasimhan [J. Algebr. Geom. 6, No. 4, 697–715 (1997; Zbl 0891.14002)].
Whereas the first edition was nearly self-contained, this would not be an apt statement in regard of the five new chapters. Due to the more advanced nature of those, the authors had to utilize several (recent) results, the proofs of which would be far beyond the scope of their already voluminous book. Nevertheless, in order to ease the reader’s effort and attain relative self-containedness, they have added four brief appendices summarizing basic facts from algebraic geometry (\(\mathbb{Q}\)-divisors, Kodaira dimension, vanishing theorems, intersection theory, and adjoint ideals), the theory of derived categories, moduli spaces of sheaves, and abelian schemes, respectively. Also, each chapter ends with a section called “Exercises and Further Results”. Actually, apart from a few true exercises, these sections mainly contain some more recent results for which the authors would have liked to give full proofs, but found neither the time nor the space to do so. Thus the reader will get to know that there is still much more to be studied, and she/he is well-guided to accomplish this additional task.
Summing up, the second, amply enlarged and up-dated edition of this outstanding standard monograph on complex abelian varieties has increased its utility in a significant degree, and its leading position among the existing books on the subject has been evidently strengthened.

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
14C15 (Equivariant) Chow groups and rings; motives
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)