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**Abelian varieties.**
*(English)*
Zbl 0223.14022

This is a remarkable book. The analytic and the algebraic theory of abelian varieties are brought together, which up to now did not happen before in one volume. The last 30 years it seems to be a sophisticated custom in algebraic geometry to hide geometric ideas and analytic motivations by wrapping them up into certainly useful, but often rather dry and complicated algebraic or functorial terminology. In this book the author uses a direct style, simplified notations and advanced methods. This sounds rather contradictory, but the result is a book which carries the reader immediately to the heart of the matter. An abelian variety is a complete irreducible group variety; the group structure is automatically commutative under these conditions; these varieties turned up naturally in the study of abelian integrals and their periods, which seems to explain the terminology. An abelian variety over the complex numbers is a complex torus, i.e. a compact connected complex Lie group, thus an analytic manifold of the form \(\mathbb{C}^g/U\), where \(U\cong \mathbb{Z}^{2g}\) is a lattice in \(\mathbb{C}^g\); conversely such an analytic manifold is an algebraic manifold, and hence an abelian variety, if and only if it can be embedded into a projective space. Abelian varieties not only turn up in analytic theories but also in algebraic number theory and in various aspects of algebraic geometry. A connected group variety is an extension of an abelian variety by a linear group; thus the study of group varieties naturally splits into two. In contrast with the theory of linear algebraic groups, for abelian varieties the group theory is easy, but the geometry is complicated. In the first chapter (in less than 40 pages) the analytic theory is developed. Special attention is paid to the computation of cohomology groups, results which further in the book will find their algebraic analogues. Anyone who has struggled through the book of F. Conforto [Abelsche Funktionen und algebraische Geometrie (Berlin) (1956; Zbl 0074.36601)], or who has studied A. Weil’ s book [Introduction á l’etude des varietes kähleriennes (Paris) (1958; Zbl 0137.41103)] will admire this penetrating approach.

The second chapter, algebraic theory via varieties, sets the first example of the conviction of the author that sheaves and their cohomology yield important tools. The fact that multiplication by \(n\) on an abelian variety of dimension \(g\) has degree \(n^{2g}\), first proved by A. Weil in the algebraic case, obvious in the analytic case, is proved here in a short way. One of the new ideas exposed in the book is a direct construction of the dual abelian variety, i.e. its Picard variety, namely by dividing out \(X\) by the kernel of the isogeny \(\varphi_L : X\to \mathrm{Pic}(X)\) defined by a polarization. In order to be able to do so, one has to prove that \(\mathrm{Ker}(\varphi_L)\) is an algebraic group (scheme), which is not difficult in characteristic zero, which is more subtile in characteristic \(p>0\) due to the fact that \(\varphi_L\) may be inseparable (and hence \(\mathrm{Ker}(\varphi_L)\) non reduced).

The third chapter, algebraic theory via schemes, centers around this technical, and certainly important point. The proof of the strengthened version of the theorem of the cube (pp. 89–93) sets a fine example in which way scheme theoretic methods supply the necessary infinitesimal techniques to match algebraic rigidity and geometric intuition. Thus direct methods prove that \(K(L)=\mathrm{Ker}(\varphi_L)\) is an algebraic group scheme, and the construction of \(\hat{x} = X/K(L)\) follows. The methods of descent are applied to give a short proof of the duality theorem: for any isogeny \(f:X\to Y\), the kernel of the dual of \(f\) is the dual of the finite group scheme \(\mathrm{Ker}(f)\). The methods of sheaf cohomology are used to prove the vanishing theorem and the Riemann Roch theorem.

The last chapter deals with the endomorphism ring: the algebraic restrictions on \(E = \operatorname{Hom}(X,X)\) coming from geometric properties of \(X\), a nice explanation of the different isogeny classes of a CM type abelian variety according to different embeddings of the field of complex multiplications into the complex numbers, and elliptic curves. The last two sections cover the theory of theta groups and the Riemann form of a line bundle (corollary: every abelian variety is isogenous to a principally polarized abelian variety) plus its relation with the analytic form. Thus the circle closes: the analytic results exposed in beginning and end of the book nicely explain and profit from their algebraic analogues. The printing and the way of displaying formulas is of high quality; we mention two misprints: an Page 196, last formula: read \(\sum\) instead of \(\prod\); an Page 217, line 5 from bottom, read “all” instead of “suitable”. The author is somewhat brief an the point of references; for two reasons it seems desirable to have more: firstly, it is not clear the reader always knows his way to the literature for concepts used but not defined in this book (a few more references would be of great help to non specialists); secondly, an several points theorems are proved, which also can be found in earlier literature; for some readers it may be easier to read earlier versions, and it seems fair to others to include references to earlier proofs of nontrivial results. We admire the style of exposition. Instead of giving explicitly in full detail all arguments which would have resulted in a doubly sized unreadable book the author phrases his proofs in such a way that verification of the arguments can be worked out, an enjoyable and stimulating task. In this way the book favourably contrasts with some modern volumes in which the real issues get lost in too many details and expanded notations. Experience has shown that a student with basic knowledge in algebraic geometry and some guidance is able to enjoy the treasures hidden in these 240 pages. There are several new results. Some of them were “well-known” but never published, several of them are contributions of the author. We mention the clear style and improvements due to the efforts of C. P. Ramanujam when writing up the notes after the course given at the Tata Institute of Fundamental Research. We strongly recommed this book, if not to every mathematician, certainly to analysts, geometers and algebraists. Getting every detail straight can be a task of several weeks, but obtaining a feeling for the many aspects of this beautiful field certainly is possible after some enjoyable hours of reading.

The second chapter, algebraic theory via varieties, sets the first example of the conviction of the author that sheaves and their cohomology yield important tools. The fact that multiplication by \(n\) on an abelian variety of dimension \(g\) has degree \(n^{2g}\), first proved by A. Weil in the algebraic case, obvious in the analytic case, is proved here in a short way. One of the new ideas exposed in the book is a direct construction of the dual abelian variety, i.e. its Picard variety, namely by dividing out \(X\) by the kernel of the isogeny \(\varphi_L : X\to \mathrm{Pic}(X)\) defined by a polarization. In order to be able to do so, one has to prove that \(\mathrm{Ker}(\varphi_L)\) is an algebraic group (scheme), which is not difficult in characteristic zero, which is more subtile in characteristic \(p>0\) due to the fact that \(\varphi_L\) may be inseparable (and hence \(\mathrm{Ker}(\varphi_L)\) non reduced).

The third chapter, algebraic theory via schemes, centers around this technical, and certainly important point. The proof of the strengthened version of the theorem of the cube (pp. 89–93) sets a fine example in which way scheme theoretic methods supply the necessary infinitesimal techniques to match algebraic rigidity and geometric intuition. Thus direct methods prove that \(K(L)=\mathrm{Ker}(\varphi_L)\) is an algebraic group scheme, and the construction of \(\hat{x} = X/K(L)\) follows. The methods of descent are applied to give a short proof of the duality theorem: for any isogeny \(f:X\to Y\), the kernel of the dual of \(f\) is the dual of the finite group scheme \(\mathrm{Ker}(f)\). The methods of sheaf cohomology are used to prove the vanishing theorem and the Riemann Roch theorem.

The last chapter deals with the endomorphism ring: the algebraic restrictions on \(E = \operatorname{Hom}(X,X)\) coming from geometric properties of \(X\), a nice explanation of the different isogeny classes of a CM type abelian variety according to different embeddings of the field of complex multiplications into the complex numbers, and elliptic curves. The last two sections cover the theory of theta groups and the Riemann form of a line bundle (corollary: every abelian variety is isogenous to a principally polarized abelian variety) plus its relation with the analytic form. Thus the circle closes: the analytic results exposed in beginning and end of the book nicely explain and profit from their algebraic analogues. The printing and the way of displaying formulas is of high quality; we mention two misprints: an Page 196, last formula: read \(\sum\) instead of \(\prod\); an Page 217, line 5 from bottom, read “all” instead of “suitable”. The author is somewhat brief an the point of references; for two reasons it seems desirable to have more: firstly, it is not clear the reader always knows his way to the literature for concepts used but not defined in this book (a few more references would be of great help to non specialists); secondly, an several points theorems are proved, which also can be found in earlier literature; for some readers it may be easier to read earlier versions, and it seems fair to others to include references to earlier proofs of nontrivial results. We admire the style of exposition. Instead of giving explicitly in full detail all arguments which would have resulted in a doubly sized unreadable book the author phrases his proofs in such a way that verification of the arguments can be worked out, an enjoyable and stimulating task. In this way the book favourably contrasts with some modern volumes in which the real issues get lost in too many details and expanded notations. Experience has shown that a student with basic knowledge in algebraic geometry and some guidance is able to enjoy the treasures hidden in these 240 pages. There are several new results. Some of them were “well-known” but never published, several of them are contributions of the author. We mention the clear style and improvements due to the efforts of C. P. Ramanujam when writing up the notes after the course given at the Tata Institute of Fundamental Research. We strongly recommed this book, if not to every mathematician, certainly to analysts, geometers and algebraists. Getting every detail straight can be a task of several weeks, but obtaining a feeling for the many aspects of this beautiful field certainly is possible after some enjoyable hours of reading.

Reviewer: Frans Oort