##
**Degeneration of abelian varieties.**
*(English)*
Zbl 0744.14031

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 22. Berlin etc.: Springer-Verlag. xii, 316 p. (1990).

This monograph provides the first systematic and comprehensive account on the theory of degenerations of abelian varieties and its application to the construction of arithmetic compactifications for the moduli spaces \(A_ g\) of principally polarized abelian varieties of dimension \(g\).

As for the construction of the moduli spaces \(A_ g\) over the integers, there are basically two methodically different approaches, namely the approach via geometric invariant theory, which is essentially due to D. Mumford [cf. Invent. Math. 1, 287-354 (1966), 3, 75-135 and 215-244 (1967; Zbl 0219.14024)] and uses symmetric ample line bundles and algebraic theta structures, and the approach via deformation theory and algebraization of formal moduli. This method has been developed, in a general context, by M. Artin about twenty years ago. In the classical case of the complex groundfield \(\mathbb{C}\), there are several compactification theories available, e.g., the Satake-Baily-Borel compactification of arithmetically defined quotients of bounded symmetric domains, or the toroidal compactification for locally symmetric varieties. However, both compactifications of \(A_ g\) (in the classical case) are not known to represent any “good” moduli functor, and this makes it difficult to interprete them algebraically and to generalize them to the general case (of arithmetical compactifications). Nevertheless, arithmetical compactifications of \(A_ g\) have been recently constructed, namely by G. Faltings in 1985 [cf. Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 321-383 (1985; Zbl 0597.14036)] and by C.-L. Chai at about the same time [cf. “Compactification of Siegel moduli schemes”, Lond. Math. Soc. Lect. Notes Ser. 107 (1985; Zbl 0578.14009)].

The present monograph now gives a refined and unified presentation of both constructions. The idea is to develop a very general theory of degenerations of abelian varieties (over an arbitrary normal base scheme), then construct the arithmetical moduli scheme \(A_ g\) for abelian varieties of dimension \(g\) by Artin’s method of algebraization of formal moduli, compactify it by toroidal methods (as in C.-L. Chai’s original approach, cited above), interprete the toroidal compactifications as natural parameter spaces for the possible periods of degenerations, and finally, construct from these data some “minimal” compactification \(A_ g^*\) of \(A_ g\) over the integers.

This is basically the plan of the work carried out in the book. Along this line the authors have arranged the material as follows:

Chapter 1 is of introductory character and gives a brief account on the basic concepts and methods used in the sequel: abelian schemes, polarizations, semi-abelian schemes and tori, deformations of abelian varieties, a review of M. Artin’s theory of algebraic stacks and formal moduli, a survey on D. Mumford’s algebraic theory of theta functions, and a reminder of the complex uniformization theory for \(A_ g\).

Chapter 2 is devoted to a general theory of degenerations of polarized abelian varieties. The methods and results developed here represent an ingenious and far-reaching generalization of the earlier work by J. Tate, D. Mumford, and M. Raynaud on semi-abelian families \(G\to S\). The authors obtain a rather complete classification in case \(S\) is the spectrum of a complete local ring, and some important new results also for general (normal) base schemes \(S\). In particular, the treatment of polarization of degree greater than 1 and the crucial case of characteristic 2 is the first published version of this fundamental improvement by the authors.

Chapter 3 deals with Mumford’s famous construction. This means, while chapter 2 was to answer the question of how degenerations of abelian varieties give rise to periods, chapter 3 just concerns the converse problem of constructing all possible degenerations. The authors follow D. Mumford’s fundamental approach [cf. D. Mumford, Compos. Math. 24, 129-174 (1972; Zbl 0228.14011)] and show that his construction gives all degenerations. In addition, the computation of the Kodaira-Spencer class of a degeneration is explicitly carried out, and that in terms of the degeneration data. — For the understanding of this chapter, Mumford’s beautiful original paper is of fundamental importance. For this reason, the authors have added a reprinted version of it, as an appendix to their book.

Chapter 4 discusses the toroidal compactification of the moduli scheme \(A_ g\). This is done in several steps, starting from the construction of so-called admissible polyhedral cone decompositions, using then Mumford’s construction to produce collections of semi-abelian degenerate families of principally polarized abelian varieties over complete rings (with respect to the combinatorial decomposition data), applying M. Artin’s method of algebraic approximation of structures over complete local rings (in order to obtain “local models” for the compactifying schemes) and, concludingly, by using the results from the previously developed general theory of degeneration, glueing together the local models to obtain an algebraic stack \(\overline A_ g\). This stack \(\overline A_ g\) is shown to be equipped with a semi-abelian scheme \(G\to\overline A_ g\) which extends the universal abelian scheme over the moduli space \(A_ g\).

In chapter 5 the authors construct an arithmetic compactification \(A^*_ g\) of \(A_ g\), which generalizes the classical Satake-Baily- Borel compactification. This is done by developing an arithmetic theory of Siegel modular forms over any toroidal compactification \(\overline A_ g\) of \(A_ g\), where \(\overline A_ g\) is the algebraic stack constructed in chapter 4. The minimal compactification \(A^*_ g\) is then obtained as the projective spectrum of the graded ring of these (generalized) Siegel modular forms. The authors also discuss the projectivity of \(\overline A_ g\), some extension theorems for semi- abelian schemes, and various other applications to arithmetical algebraic geometry. — The methods developed in chapters 4 and 5 are shown to be applicable also to abelian varieties with level-\(n\) structures and their moduli spaces \(A_{g,n}\), and they can be used to investigate Shimura varieties more closely.

In chapter 6 the equivariant sheaves on \(A_ g\) and \(\overline A_ g\) are studied. They appear as vector bundles, or crystals, or étale sheaves. Their interrelation with the representation theory of certain algebraic groups, with Deligne’s Hodge theory, with De Rham cohomology, and with the crystalline cohomology is discussed in great detail. The main results of this chapter concern the Betti cohomology of \(A_ g(\mathbb{C})\) and its Hodge structure. They are based upon G. Faltings’ previous approach by using Bernstein-Gelfand-Gelfand resolutions [cf. G. Faltings in Sémin. d’Algèbre, Dubreil-Malliavin, 35ème Année, Proc. Paris 1982, Lect. Notes Math. 1029, 55-98 (1983; Zbl 0539.22008)], but extend those ones significantly.

The final chapter 7 discusses the action of Hecke operators on all the sheaves and their cohomologies that have been studied in chapter 6. This topic is still far from being completely understood, and therefore the authors limits themselves to give the basic concepts, developed methods, open problems, and possible applications to the further study of the arithmetic of Siegel modular forms. In the course of this discussion, however, a proof of D. Deligne’s recent result on the equation for the Frobenius over the Hecke algebra is given.

Altogether, this monograph leads the reader to the forefront of current research in algebraic geometry, especially in arithmetic algebraic geometry. — It is certainly an outstanding book, because it contains a wealth of ideas, methods and results that have never been published in such deep-going, systematic and comprehensive a way. Many results are even published for the first time, in this full generality. The reader is required to have a very profound knowledge in algebraic geometry, arithmetic, and related branches in mathematics. Thus the text is not easily accessible for non-specialists in the field, but it is of invaluable importance for active researchers, interested graduate students, and mathematically advanced physicists working with algebro- geometric methods in conformal quantum field theory. For those readers, this book may serve as both a textbook/reference and a source for further research. In this regard, the authors have succeeded to put a milestone within the development of this extremely difficult topic, and a point of culmination in mathematical culture.

As for the construction of the moduli spaces \(A_ g\) over the integers, there are basically two methodically different approaches, namely the approach via geometric invariant theory, which is essentially due to D. Mumford [cf. Invent. Math. 1, 287-354 (1966), 3, 75-135 and 215-244 (1967; Zbl 0219.14024)] and uses symmetric ample line bundles and algebraic theta structures, and the approach via deformation theory and algebraization of formal moduli. This method has been developed, in a general context, by M. Artin about twenty years ago. In the classical case of the complex groundfield \(\mathbb{C}\), there are several compactification theories available, e.g., the Satake-Baily-Borel compactification of arithmetically defined quotients of bounded symmetric domains, or the toroidal compactification for locally symmetric varieties. However, both compactifications of \(A_ g\) (in the classical case) are not known to represent any “good” moduli functor, and this makes it difficult to interprete them algebraically and to generalize them to the general case (of arithmetical compactifications). Nevertheless, arithmetical compactifications of \(A_ g\) have been recently constructed, namely by G. Faltings in 1985 [cf. Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 321-383 (1985; Zbl 0597.14036)] and by C.-L. Chai at about the same time [cf. “Compactification of Siegel moduli schemes”, Lond. Math. Soc. Lect. Notes Ser. 107 (1985; Zbl 0578.14009)].

The present monograph now gives a refined and unified presentation of both constructions. The idea is to develop a very general theory of degenerations of abelian varieties (over an arbitrary normal base scheme), then construct the arithmetical moduli scheme \(A_ g\) for abelian varieties of dimension \(g\) by Artin’s method of algebraization of formal moduli, compactify it by toroidal methods (as in C.-L. Chai’s original approach, cited above), interprete the toroidal compactifications as natural parameter spaces for the possible periods of degenerations, and finally, construct from these data some “minimal” compactification \(A_ g^*\) of \(A_ g\) over the integers.

This is basically the plan of the work carried out in the book. Along this line the authors have arranged the material as follows:

Chapter 1 is of introductory character and gives a brief account on the basic concepts and methods used in the sequel: abelian schemes, polarizations, semi-abelian schemes and tori, deformations of abelian varieties, a review of M. Artin’s theory of algebraic stacks and formal moduli, a survey on D. Mumford’s algebraic theory of theta functions, and a reminder of the complex uniformization theory for \(A_ g\).

Chapter 2 is devoted to a general theory of degenerations of polarized abelian varieties. The methods and results developed here represent an ingenious and far-reaching generalization of the earlier work by J. Tate, D. Mumford, and M. Raynaud on semi-abelian families \(G\to S\). The authors obtain a rather complete classification in case \(S\) is the spectrum of a complete local ring, and some important new results also for general (normal) base schemes \(S\). In particular, the treatment of polarization of degree greater than 1 and the crucial case of characteristic 2 is the first published version of this fundamental improvement by the authors.

Chapter 3 deals with Mumford’s famous construction. This means, while chapter 2 was to answer the question of how degenerations of abelian varieties give rise to periods, chapter 3 just concerns the converse problem of constructing all possible degenerations. The authors follow D. Mumford’s fundamental approach [cf. D. Mumford, Compos. Math. 24, 129-174 (1972; Zbl 0228.14011)] and show that his construction gives all degenerations. In addition, the computation of the Kodaira-Spencer class of a degeneration is explicitly carried out, and that in terms of the degeneration data. — For the understanding of this chapter, Mumford’s beautiful original paper is of fundamental importance. For this reason, the authors have added a reprinted version of it, as an appendix to their book.

Chapter 4 discusses the toroidal compactification of the moduli scheme \(A_ g\). This is done in several steps, starting from the construction of so-called admissible polyhedral cone decompositions, using then Mumford’s construction to produce collections of semi-abelian degenerate families of principally polarized abelian varieties over complete rings (with respect to the combinatorial decomposition data), applying M. Artin’s method of algebraic approximation of structures over complete local rings (in order to obtain “local models” for the compactifying schemes) and, concludingly, by using the results from the previously developed general theory of degeneration, glueing together the local models to obtain an algebraic stack \(\overline A_ g\). This stack \(\overline A_ g\) is shown to be equipped with a semi-abelian scheme \(G\to\overline A_ g\) which extends the universal abelian scheme over the moduli space \(A_ g\).

In chapter 5 the authors construct an arithmetic compactification \(A^*_ g\) of \(A_ g\), which generalizes the classical Satake-Baily- Borel compactification. This is done by developing an arithmetic theory of Siegel modular forms over any toroidal compactification \(\overline A_ g\) of \(A_ g\), where \(\overline A_ g\) is the algebraic stack constructed in chapter 4. The minimal compactification \(A^*_ g\) is then obtained as the projective spectrum of the graded ring of these (generalized) Siegel modular forms. The authors also discuss the projectivity of \(\overline A_ g\), some extension theorems for semi- abelian schemes, and various other applications to arithmetical algebraic geometry. — The methods developed in chapters 4 and 5 are shown to be applicable also to abelian varieties with level-\(n\) structures and their moduli spaces \(A_{g,n}\), and they can be used to investigate Shimura varieties more closely.

In chapter 6 the equivariant sheaves on \(A_ g\) and \(\overline A_ g\) are studied. They appear as vector bundles, or crystals, or étale sheaves. Their interrelation with the representation theory of certain algebraic groups, with Deligne’s Hodge theory, with De Rham cohomology, and with the crystalline cohomology is discussed in great detail. The main results of this chapter concern the Betti cohomology of \(A_ g(\mathbb{C})\) and its Hodge structure. They are based upon G. Faltings’ previous approach by using Bernstein-Gelfand-Gelfand resolutions [cf. G. Faltings in Sémin. d’Algèbre, Dubreil-Malliavin, 35ème Année, Proc. Paris 1982, Lect. Notes Math. 1029, 55-98 (1983; Zbl 0539.22008)], but extend those ones significantly.

The final chapter 7 discusses the action of Hecke operators on all the sheaves and their cohomologies that have been studied in chapter 6. This topic is still far from being completely understood, and therefore the authors limits themselves to give the basic concepts, developed methods, open problems, and possible applications to the further study of the arithmetic of Siegel modular forms. In the course of this discussion, however, a proof of D. Deligne’s recent result on the equation for the Frobenius over the Hecke algebra is given.

Altogether, this monograph leads the reader to the forefront of current research in algebraic geometry, especially in arithmetic algebraic geometry. — It is certainly an outstanding book, because it contains a wealth of ideas, methods and results that have never been published in such deep-going, systematic and comprehensive a way. Many results are even published for the first time, in this full generality. The reader is required to have a very profound knowledge in algebraic geometry, arithmetic, and related branches in mathematics. Thus the text is not easily accessible for non-specialists in the field, but it is of invaluable importance for active researchers, interested graduate students, and mathematically advanced physicists working with algebro- geometric methods in conformal quantum field theory. For those readers, this book may serve as both a textbook/reference and a source for further research. In this regard, the authors have succeeded to put a milestone within the development of this extremely difficult topic, and a point of culmination in mathematical culture.

Reviewer: W.Kleinert (Berlin)

### MSC:

14K10 | Algebraic moduli of abelian varieties, classification |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14K15 | Arithmetic ground fields for abelian varieties |

14L15 | Group schemes |

14D10 | Arithmetic ground fields (finite, local, global) and families or fibrations |

14D15 | Formal methods and deformations in algebraic geometry |

14L05 | Formal groups, \(p\)-divisible groups |