This book provides a very modern introduction to the theory of abelian varieties and their theta functions, with a special emphasis on those new aspects that have vigorously pervaded this classic subject over the last two decades. Namely, in 1981, S. Mukai discovered a striking algebro-geometric analogue of the Fourier transform in the context of abelian varieties [cf.S. Mukai, Nagoya Math. J. 81, 153-175 (1981;Zbl 0417.14036)], which is now called the “Fourier-Mukai transform”, and one of the main goals of the present book is to present an introduction to the algebraic theory of abelian varieties in which this new and important concept takes its proper place, very much so for the first time in the existing textbook literature on abelian varieties and theta functions. On the one hand, the use of the Fourier-Mukai transform leads to more conceptual proofs of many classic theorems, and offers another view to the theory as a whole. On the other hand, the analogy with the original Fourier transform in analysis inaugurates new directions in the study of abelian varieties. Finally, another strong motivation for presenting the theory of abelian varieties in the context of the categorical framework of the Fourier-Mukai transform is given by the recently discovered links to the mirror symmetry problem in algebraic geometry and quantum field theory.
It is therefore more than rewarding, and deserving of thanks, that the author, who is one of the leading experts in these fields of current research, has presented this new, largely up-to-date introduction to abelian varieties and theta functions to the mathematical community. The text is based on lectures delivered by the author at Harvard University (1998) and Boston University (2001). It is primarily intended for seasoned graduate students and postgraduate researchers interested in modern algebraic geometry and mathematical physics. The main idea that influenced the structure of this modern textbook is, in contrast to the earlier great standard texts on (complex) abelian varieties and theta functions (J. Igusa, D. Mumford, C. Birkenhake and H. Lange, G. Kempf, A. Weil, and others), the more recent idea of “categorification” in mathematics, that is the process of finding category-theoretic analogues of set-theoretic concepts, methods, and techniques à la Mukai, Kontsevich, Khovanov, Fukaya, and others. This approach has led to new, deep and spectacular insights and results, over the past twenty years, and the text under review reflects this contemporary (and extremely successful) trend in an opportune way. As to the contents, the book is subdivided into three main parts, each of which comes with seven or eight separate chapters. Part I is entitled “Analytic Theory” and discusses the transcendental theory of abelian varieties, and that in both its classic and more recent aspects.
Chapter 1 provides the well-known classification of holomorphic line bundles on complex tori, while in Chapters 2-5 the main focus is on the theory of theta functions. The author’s approach is heavily based on the representation theory of the Heisenberg group, rather than on the geometry of theta functions, and culminates in the study of the functional equation for theta functions. Chapters 6 and 7 present some very recent material, namely the study of mirror symmetry between symplectic tori and complex tori by means of Lagrangian tori fibrations. At the end of Part I, in Chapter 7, this is used to compute the cohomology of line bundles on complex tori via the mirror symmetry approach.
Part II comes with the title “Algebraic Theory” and incorporates the following eight chapters. The objects of study are here the general abelian varieties over an algebraically closed field of arbitrary characteristic. Chapters 8-10 are devoted to the classic foundational material: abelian varieties over arbitrary fields, the theorem of the cube, the dual abelian variety extensions and biextensions of abelian varieties, and duality theory. The author gives here a nicely condensed account of the rich theory which, in its full depth, can be found in D. Mumford’s unrivalled book “Abelian Varieties” (Oxford University Press, 1974). Then, in Chapter 11, the Fourier-Mukai transform is introduced via functors between derived categories and coherent sheaves. Using this transform, the author gives another, more conceptual proof of the main theorem about duality of abelian varieties and, furthermore, a new view to the cohomology of nondegenerate line bundles on abelian varieties.
Chapter 12 provides an algebraic analogue of the representation theory of the Heisenberg group in terms of the Mumford group, together with an application to the proof of Riemann’s quartic theta identity. Chapter 13 is devoted to the study of symmetric line bundles on arbitrary abelian varieties, whereas in Chapter 14 the foregoing approach via the Fourier-Mukai transform is used to recover Atiyah’s classification of vector bundles over elliptic curves.
In the final Chapter 15 of this part of the book, the author develops a “categorification” of the representation theory of Heisenberg groups, in which the Fourier-Mukai transform replaces the usual Fourier transform. The main result is a construction of equivalences between derived categories of coherent sheaves on abelian varieties, which puts the construction of the intertwining operators in the classical theory into a general categorical framework.
Part III is entitled “Jacobians” and contains the remaining seven chapters of the book. The theory of Jacobian varieties of smooth irreducible projective curves over arbitrary fields, which is scarcely covered by the other textbooks on general abelian varieties, is here discussed in greater detail.
Chapter 16 deals with the construction of the Jacobian of a curve by means of symmetric products of curves, including some basic results on symmetric powers that work in any characteristic.
Chapter 17 explains the principal polarization on the Jacobian and gives a modern treatment of some classical topics such as determinant bundles, embedding of a curve into its Jacobian, the geometry of the theta divisor, theta identities, the Albanese variety, and theta characteristics. Fay’s trisecant formula and a general proof of it constitute the topic of Chapter 18. The author combines here the foregoing results from Chapter 17 with the residue calculus for rational differentiale on a curve, together with a study of the so-called Cauchy-Szegö kernels associated to certain line bundles on the curve.
Chapter 19 presents a more detailed study of the symmetric powers of a curve, culminating in the calculation of Picard groups and in the cohomological vanishing theorem for some natural vector bundles on the symmetric powers of a curve. Some Brill-Noether theory is discussed in Chapter 20, where the varieties of special divisors on a curve, estimates on their dimensions and an explicit description of the tangent cones to their singular points are the main topics.
Chapter 21 gives a treatment of the Torelli theorem for curves, together with a new proof using the Fourier-Mukai transform. Finally, in Chapter 22, the author turns to some very recent developments, namely the Deligne symbol for a pair of line bundles on a relative curve, generalized theta divisors, and the so-called strange duality conjecture generalizing theta functions to moduli spaces of vector bundles on curves. The novelty is here a reformulation of this conjecture in a symmetric way by using the Fourier-Mukai transform.
Each chapter comes with a bunch of exercises complementing the material covered by the text. These exercises, varying in their degree of difficulty and depth, are mostly quite challenging. However, they are perfectly suited to guide the reader into further domains, and even to the forefront of current research, though the hints for solution are quite sparse. On the other hand, there is a carefully prepared list of bibliographical notes and hints for further reading at the end of the book, which should by very useful to the reader. The author has given rather detailed comments and hints for each single chapter together with many valuable instructions for additional reading. Also, he is fair and honest enough to tell the reader where some of the proofs given in the text are borrowed from. Needless to say, the bibliography is very comprehensive (138 titles) and up-to-date.
All together, this text is a highly welcome and valuable enhancement of the existing literature in the field. Apart from covering new grounds, the author explains some of the more recent ideas and perspectives in the theory of abelian varieties and theta functions with great expertise. The exposition captivates by its systematic clarity, indicated profundity, necessary rigor, and masterly conciseness. Of course, and as the author himself emphasizes in the introduction to the book, this text cannot be seen as a universal improvement of all the other (excellent) accounts on abelian varieties and theta functions, but it certainly represents a fresh breeze as well as a touch of future in this evergreen field of mathematical research. However, this is not a primer for beginners in this area. The reader is required to have a good deal of basic knowledge in complex and differential geometry, classical Fourier analysis, representation theory, modern algebraic geometry, and categorical algebra. At any rate, this book will rank among the most important monographs on abelian varieties and theta functions.

Reviewer: Werner Kleinert (Berlin)