Complex Abelian Varieties

Abstract

This is the second, substantially extended, edition of the book first published in 1994. The major change from the previous edition is that five new chapters (Chapters 13-17) and four appendices (Appendices C-F) have been added. Since an excellent review by Debarre for the first edition Complex abelian varieties, Springer, Berlin, 1992; MR1217487 (94j:14001) already exists, the present review is focusing mainly on the newly added parts. Chapter 13 deals with automorphisms of abelian varieties. The first two sections show how one makes use of the fixed-point sets of an automorphism to obtain restrictions on the possible automorphism groups. Section 13.3 discusses abelian varieties of CM-type, and determines all abelian varieties admitting an automorphism with finite fixed-point set, whose number of eigenvalues is minimal. The remaining sections are concerned with the case of abelian surfaces, and an extension of Poincar reducibility for abelian varieties with automorphism groups. Chapter 14 deals with vector bundles on abelian varieties, focusing on the theory of Fourier transform due to S. Mukai Nagoya Math. J. 81 (1981), 153-175; MR0607081 (82f:14036). The authors take a didactical point of view, showing that for some sheaves, called WIT-sheaves, one can define the Fourier transform on the level of sheaves without having to go to the derived category. The first four sections contain the definition and some properties of the Fourier transform of WIT-sheaves. In Sections 14.5 and 14.6 some applications on global generation and Picard sheaves are discussed. Section 14.7 contains the general definition of the Fourier transform of a complex. In the final section vector bundles on abelian surfaces are considered. The definitions and results on derived categories needed in this chapter are complied in Appendix D, and those on moduli spaces of sheaves are in Appendix E. Chapter 15 presents recent results on line bundles and the theta divisor, especially on syzygies, Seshadri constants of an ample line bundle, and the singularities of the theta divisor. These results rely on some deep theorems, such as Kawamata-Viehweg theorem, in birational geometry, some definitions and properties of which are compiled in Appendix C. Chapter 16 deals with cycles on abelian varieties. This chapter is made as self-contained as possible by the first two sections where the main properties of the Chow ring and correspondences are introduced. Sections 16.3 and 16.4 define the Fourier transforms on the Chow ring and the cohomology ring. As an application, Section 16.5 gives an eigenspace decomposition of the Chow ring induced by the multiplication map. As is explained in Appendix F, this result can be generalized to abelian schemes, and the latter yields a Kunneth decomposition of the Chow ring of the self-product of abelian varieties, which is the main content of Section 16.6. Section 16.7 gives a short and new proof of results of Bloch on zero cycles on abelian varieties. In Chapter 17 the authors give an introduction to the Hodge conjecture for abelian varieties, and outline the proof for general abelian and Jacobian varieties. The first two sections introduce Hodge structures and study their relation to complex structure. In Section 17.3 they introduce the Hodge group of an abelian variety and give complete proofs for most of its fundamental properties. The final two sections contain the proof of the Hodge conjecture for general abelian and Jacobian varieties. As is the case for the first 12 chapters and two appendices, the exposition in the new chapters and appendices is very clear. The book well deserves to become a standard reference for more researchers working or interested in the theory of abelian varieties.