There are three articles on abelian varieties; the first by Rosen is on the analytic theory, while the other two by Milne are on the geometric theory in arbitrary characteristics and on Jacobian varieties, respectively. These are the three main approaches to the theory of abelian varieties, and to have all three represented in one place is very pleasant. For example, at the elementary level, the reader can compare the proof that a connected compact complex Lie group is commutative, in Rosen's article, with the proof that a complete group variety is commutative, in the article "Abelian varieties" by Milne. (Note that Section 16 describes Zarkhin's trick, which is used in Faltings' paper.) The article "Jacobian varieties " by Milne is a modern treatment of the subject, and as such helps fill an important gap in the expository literature, since \n D. Mumford\en never wrote the second volume of his book Abelian varieties \ref[Oxford Univ. Press, London, 1970; MR 44 #219], and the book Abelian varieties by \n S. Lang\en \ref[Interscience, New York, 1959; MR 21 #4959] was written in the language of Weil. The bibliographical notes to Milne's article are well worth reading; it is a pity that this kind of thing is not done more often.

Reviewed by H. Gillet