Moduli spaces of abelian surfaces: compactification, degenerations, and theta functions.

*(English)*Zbl 0809.14035
De Gruyter Expositions in Mathematics. 12. Berlin: Walter de Gruyter. xii, 347 p. (1993).

This book deals with the problem of constructing compactifications of moduli spaces of abelian varieties. The problem has been attacked by many people and in many different ways.

The first solution was given by Satake. The disadvantage of his construction is the fact that the boundary of his compactification has many singularities. Another way of attacking the problem is Mumford’s very general method of toroidal compactifications of quotients of bounded symmetric spaces. A special case of this method gives Igusa’s compactification. – In this book the authors use Mumford’s method to construct a compactification of the moduli space \({\mathcal A} (1,p)\) of abelian surfaces with a polarization of type \((1,p)\) where \(p\) is an odd prime or \(p=1\).

The book has three main parts. Part I gives a description of the space \({\mathcal A} (1,p)\) and its compactification \({\mathcal A}^* (1,p)\). The Igusa compactification corresponds with the special case \(p=1\). – In the chapters 1 and 2 a description of the moduli problem and of Mumford’s theory of torus embeddings is given. In chapter 3 the authors give a detailed description of the toroidal compactification \({\mathcal A}^* (1,p)\) and its boundary. – A good compactification of a moduli space of abelian varieties should have the property that boundary points correspond with degenerations of the objects parametrized by the given moduli space. In general toroidal compactifications do not have this property.

The aim of part II is to associate to each boundary point of \({\mathcal A}^* (1,p)\) a degenerated abelian surface. The basic tool is Mumford’s construction of degenerating abelian varieties over complete local rings. In particular the “final example” of Mumford’s paper will be important for this correspondence. In the chapters 1 and 2 Mumford’s theory and his “final example” are recalled. In chapter 3 a degenerated abelian surface is constructed for each point of \({\mathcal A}^* (1,1) \backslash {\mathcal A} (1,1)\). The results are then applied to obtain degenerated abelian surface for the boundary points of \({\mathcal A}^* (1,p)\).

In part III the special case of \({\mathcal A}^* (1,5)\) and its relation with the Horrocks-Mumford bundle \(F\) on \(\mathbb{P}^ 4\) is studied. If the zero-set \(X_ s\) of a nonzero section \(s\) of the Horrocks-Mumford bundle \(F\) is smooth then \(X_ s\) is an abelian surface which has a canonical polarization of type (1,5). This observation leads to an isomorphism of a Zariski open subset in the projective space \(\mathbb{P} \Gamma\) of sections of \(F\) to an open subset of the moduli space \({\mathcal A} (1,5)\). An extension of the inverse of this isomorphism is constructed as a birational correspondence from \({\mathcal A}^* (1,p)\) to \(\mathbb{P} \Gamma\).

The first solution was given by Satake. The disadvantage of his construction is the fact that the boundary of his compactification has many singularities. Another way of attacking the problem is Mumford’s very general method of toroidal compactifications of quotients of bounded symmetric spaces. A special case of this method gives Igusa’s compactification. – In this book the authors use Mumford’s method to construct a compactification of the moduli space \({\mathcal A} (1,p)\) of abelian surfaces with a polarization of type \((1,p)\) where \(p\) is an odd prime or \(p=1\).

The book has three main parts. Part I gives a description of the space \({\mathcal A} (1,p)\) and its compactification \({\mathcal A}^* (1,p)\). The Igusa compactification corresponds with the special case \(p=1\). – In the chapters 1 and 2 a description of the moduli problem and of Mumford’s theory of torus embeddings is given. In chapter 3 the authors give a detailed description of the toroidal compactification \({\mathcal A}^* (1,p)\) and its boundary. – A good compactification of a moduli space of abelian varieties should have the property that boundary points correspond with degenerations of the objects parametrized by the given moduli space. In general toroidal compactifications do not have this property.

The aim of part II is to associate to each boundary point of \({\mathcal A}^* (1,p)\) a degenerated abelian surface. The basic tool is Mumford’s construction of degenerating abelian varieties over complete local rings. In particular the “final example” of Mumford’s paper will be important for this correspondence. In the chapters 1 and 2 Mumford’s theory and his “final example” are recalled. In chapter 3 a degenerated abelian surface is constructed for each point of \({\mathcal A}^* (1,1) \backslash {\mathcal A} (1,1)\). The results are then applied to obtain degenerated abelian surface for the boundary points of \({\mathcal A}^* (1,p)\).

In part III the special case of \({\mathcal A}^* (1,5)\) and its relation with the Horrocks-Mumford bundle \(F\) on \(\mathbb{P}^ 4\) is studied. If the zero-set \(X_ s\) of a nonzero section \(s\) of the Horrocks-Mumford bundle \(F\) is smooth then \(X_ s\) is an abelian surface which has a canonical polarization of type (1,5). This observation leads to an isomorphism of a Zariski open subset in the projective space \(\mathbb{P} \Gamma\) of sections of \(F\) to an open subset of the moduli space \({\mathcal A} (1,5)\). An extension of the inverse of this isomorphism is constructed as a birational correspondence from \({\mathcal A}^* (1,p)\) to \(\mathbb{P} \Gamma\).

Reviewer: G.Van Steen (Antwerpen)

##### MSC:

14K10 | Algebraic moduli of abelian varieties, classification |

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

32J05 | Compactification of analytic spaces |

14K25 | Theta functions and abelian varieties |