This book provides a modern, i.e., scheme-theoretic, treatment of most of the basic theory of abelian varieties.
Chapter 1 is devoted to the analytic theory of abelian varieties over the complex numbers. It is first shown that any complex compact connected Lie group $X$ is a torus, and then the various cohomology groups $H^q(X, Z)$, $H^q(X, C)$, $H^q(X,\Omega^p)$ are computed and related. Line bundles on $X$ are classified in terms of Hermitian forms and functions (Theorem of Appell-Humbert) and their sections described as theta functions. Finally, the algebra-izability of tori is investigated.

The second chapter contains elegant proofs of the seesaw theorem and the theorems of the cube and square and their consequences. The dual abelian variety (in characteristic zero) and quotients of abelian varieties by finite groups are constructed.

In Chapter 3 some of the results of Chapter 2 are extended to algebraic schemes that are not necessarily integral. Then, after a useful discussion of commutative group schemes, the quotient of a scheme by a finite group scheme is constructed. The dual abelian variety (any characteristic) is constructed and the duality theorem is proved. Finally (a highlight!), the cohomology of line bundles is treated. The Riemann-Roch and vanishing theorems are proved, a method of computing the index of a line bundle is demonstrated, and it is shown that the cube of any ample line bundle is very ample.

In the final chapter, the possible structures of the endomorphism algebra and Néron-Severi group are demonstrated, the Riemann hypothesis is proved, and the Riemann form of a divisor is constructed, once in the standard way and once by making use of the author's important concept of a theta group.

If the content of this book is compared with that of its immediate predecessor [S. Lang, Abelian varieties, Inter-science, New York, 1959; MR 21 #4959] then the most obvious and serious omission is seen to be that of any mention of Jacobian or Albanese varieties. Less serious is the omission of Chow's theory of the $K/k$ trace and image of an abelian variety. On the other hand, Mumford's book contains many results that were unknown or unproved when Lang's book was written (the Riemann-Roch theorem, the duality theorem, the theory involving finite subgroup schemes in characteristic $p$, etc.), it gives a much more complete treatment of the theory in Chapter 4, and it contains the analytic theory, which Lang omits entirely. Indeed, an outstanding feature of Mumford's book is the way in which the easier analytic theory is used to motivate and illustrate the algebraic theory throughout the book. One disappointing feature is the lack of concern with rationality questions---all ground fields are algebraically closed---which will limit the usefulness of the book, at least as a reference, for arithmetic geometers.

More striking than the difference in content of the two books is the difference in the methods used. In Mumford's book divisors and intersection theory become almost completely replaced by sheaves and cohomology. Opinions will differ on this, but the reviewer's is that the resultant gain in clarity and power far outweighs any loss in geometric intuition.

The exposition is clear, and often elegant and original. The book gives the impression that the author began with an idea of the most important results and set about proving them in the cleanest and quickest way (a method of exposition which could be recommended to other authors). There are aspects of the book one could niggle at (the lack of an index, the paucity of references) but one's final impression is of a beautiful book that will at last allow everyone to learn about abelian varieties without first having to grapple with Weil's language of algebraic geometry.

Reviewed by J. S. Milne