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Abelian varieties. With appendices by C. P. Ramanujam and Yuri Manin. Corrected reprint of the 2nd ed. 1974. (English) Zbl 1177.14001
New Delhi: Hindustan Book Agency/distrib. by American Mathematical Society (AMS); Bombay: Tata Institute of Fundamental Research (ISBN 978-81-85931-86-9/hbk). xii, 263 p. (2008).
Presented annually by the American Mathematical Society, the Leroy P. Steele Prize for Mathematical Exposition is one of the highest distinctions in Mathematics. In this category, the 2007 AMS Steele Prize was awarded to David B. Mumford in recognition of his pioneering and beautiful accounts of a host of aspects of algebraic geometry, published over a period of more than two decades. The prize citation emphasizes that, particularly in D. Mumford “The red book of varieties and schemes” (1974; Zbl 0945.14001); includes the Michigan lectures (1974) on “Curves and their Jacobians”. 2nd, expanded ed. with contributions by Enrico Arbarello. Lect. Notes Math. 1358. Berlin: Springer 306 (1999; Zbl 0223.14022)] the classical heritage is beautifully intertwined with the modern aspects of algebraic geometry, in a masterful way that sharply illuminates both. Recognizing also the series of Mumford’s other influential books published between 1965 and 1991, the prize citation concludes: “All of these books are, and will remain for the foreseeable future, classics to which the reader returns over and over”.
The book under review is the reprint of one of these distinguished expository writings of D. Mumford, namely a corrected, completely re-typeset printing of his particularly emphasized classic “Abelian Varieties”.
Based on series of lectures delivered in the winter term 1967–1968 at the Tata Institute of Fundamental Research (Mumbai/Bombay, India) and written up by the late C. P. Ramanujam, the first edition of this book appeared in 1970. The second edition, with two appendices added by C. P. Ramanujam and by Yu. I. Manin, was brought out in 1974 (Zbl 0326.14012), and the first reprint of this enlarged second edition was made available in 1985, that is, more than 20 years ago (Zbl 0583.14015).
Now, in response to continuing strong demand, the Tata Institute of Fundamental research has finally provided another reprint of the second edition of Mumford’s classic “Abelian Varieties”. In the current new reprint, the well-tried text has been left entirely intact. However, some appropriate corrections have been supplied by B. Conrad and by Ching-Li Chai, and the book appears now in modern LaTeX typesetting, with the original page numbers indicated in the margins of the present edition.
Back in 1970, Mumford’s book provided the first modern treatment of abelian varieties. Without approaching the subject by using “the crutch of Jacobian or Albanese varieties”, as it was still done in S. Lang’s book “Abelian Varieties” (1959; Zbl 0098.13201) from 1959, Mumford systematically developed the basic theory of abelian varieties both in its general abstract setting and in the framework of algebraic schemes and sheaves à la A. Grothendieck and P. Cartier, thereby clarifying the picture of the situation also in positive characteristics. Moreover, in Mumford’s book, both the analytic and the algebraic theory of abelian varieties were treated simultaneously for the first time, and for the last time so far, which represented another novelty of propelling impact.
As D. Mumford himself pointed out in the preface, this book covers roughly half of the material that would be appropriate for a reasonably complete treatment of the theory of abelian varieties. Such a discussion should also include Jacobians, abelian schemes, deformation theory and moduli problems, modular forms and the global structure of moduli spaces, Dieudonné, theory in characteristic $$p$$, rationality questions, and the arithmetic theory of abelian schemes. However, such a comprehensive text on abelian varieties still has to be written, even almost forty years after the appearance of Mumford’s modern introductory text on the subject. Therefore, apart from the various recent excellent textbooks on complex tori and complex abelian varieties (Birkenhake-Lange, Kempf, Debarre, Polishchuk, and others) and from J. S. Milne’s recent, just as excellent lecture notes “Abelian Varieties” (version of March 16, 2008, available from the author’s web site http://www.jmilne.org/math/CourseNotes/math731.html), where many of the further-leading algebraic aspects are touched upon, Mumford’s classic is now as before the definite standard text for entering the modern theory of abelian varieties. Studying Mumford’s unique book means learning directly from one of the great masters in the field, who was awarded a Fields Medal in 1974 for his epoch-making work on algebraic curves, abelian varieties, and their classification theory. It is Mumford’s mastery of expository writing, which stands out by its unequalled clarity, elegance, originality and depth, that makes all his books and papers timeless classics, in particular his masterpiece “Abelian Varieties”.
As the current new edition of this book is an unaltered reprint of the second edition from 1974, we may refer to the review of the latter D. Mumford Abelian varieties. With appendices by C. P. Ramanujam and Yuri Manin. Tata Institute of Fundamental Research Studies in Mathematics. 5. London: Oxford University Press. X, 279 p. £2.95 (1974; Zbl 0326.14012)] respecting the detailed contents. However, after so many decades, it might be worthwile to recall that the text comprises four chapters and two appendices discussing the following topics:
Chapter I: Analytic Theory (complex tori, line bundles, and algebraizabilty of tori);
Chapter II: Algebraic theory via varieties (abelian varieties, cohomology and base change, the theorem of the cube, quotients of varieties by finite groups, the dual abelian variety in characteristic 0, and the complex case);
Chapter III: Algebraic theory via schemes (basic theory of group schemes, quotients by finite group schemes, the general dual abelian variety, duality for finite commutative group schemes, applications to abelian varieties, cohomology of line bundles, and very ample line bundles);
Chapter IV: Endomorphisms of Abelian Varieties and $$\ell$$-adic Representations (étale coverings, structure of endomorphism rings of abelian varieties, Riemann forms, the Rosati involution, and Riemann forms via theta groups);
Appendix I (by C. P. Ramanujam): The Theorem of Tate;
Appendix II (by Yu. I. Manin): The Mordell-Weil theorem.
All in all, it is more than rewarding that D. Mumford’s classic standard text “Abelian Varieties” has been made available again for new generations of students, teachers, and researchers in the field. In its modern typesetting and in its corrected version, this jewel in the mathematical literature has become even more attractive and valuable.

##### MSC:
 14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14Kxx Abelian varieties and schemes 14G05 Rational points 14K05 Algebraic theory of abelian varieties 14K20 Analytic theory of abelian varieties; abelian integrals and differentials
##### MathOverflow Questions:
Do we have Hodge symmetry for char $p$?