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Abelian Varieties over the Complex Numbers

Herbert Lange
Publication Date: 
Number of Pages: 
Grundlehren Text Editions
[Reviewed by
Felipe Zaldivar
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Abel’s far-reaching generalization for hyperelliptic integrals (Abelian integrals) of Euler’s addition theorem for elliptic integrals (a finite sum of such integrals may be written as g such integrals, where g is the genus of the corresponding algebraic curve, save some logarithmic integrals), still resonates throughout the history of our science. The geometry and arithmetic of abelian varieties is a very active area of research in our days.
Essential modern monographs in this area include the one by D. Mumford, Abelian Varieties and the book by C. Birkenhake and H. Lange, Complex Abelian Varieties. For a graduate student entering this area, Mumford’s monograph is still a requirement, and Birkenhake and H. Lange’s gives a modern and comprehensive treatment of complex abelian varieties, including topics not included in Mumford’s classical text.
However, neither of these two monographs was conceived as a proper textbook, and the book under review fills this gap. The author has adapted and rewritten some chapters of the original monograph, sometimes expanding or simplifying the presentation to make it more accessible and adding relevant exercises calibrated to a beginning reader.
The reorganization of the topics is fine surgical work. Several portions of the original monograph are sewn in a natural way in the new book, adding examples or additional text when necessary, and re-arranging the focus to make it a more friendly introduction to the subject. Careful attention to details and the required background makes the book under review accessible to an interested reader and could be a used as textbook for a course on abelian varieties.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is