Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions.

The second edition contains five new chapters which present some of the most important recent result on the subject. Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture.

From the reviews:

" the authors have somehow managed to make it unique in several respects: not only is it far more readable than most of other book on the subject, but it is also much more complete. It is, in my opinion, a very valuable reference book . [] prerequisites are kept to a minimum. Very little background in algebraic geometry is necessary, almost all proofs are complete and accessible references are provided whenever they are not. Olivier Debarre in Mathematical Reviews, 1994

"It is a great reference and textbook, detailed, very up-to-date, thorough, clearly written and perfectly arranged."
W. Kleinert in Zentralblatt MATH, 1993

" written in an understandable and systematical way and can be recommended to all mathematicians and physicists interested in the subject."
Newsletter of the European Mathematical Society, 1993

From the reviews of the second edition:

"This book aims to be a course in Lie groups that can be covered in one year with a group of seasoned graduate students. offers a wealth of complementary, partly quite recent material that is not found in any other textbook on Lie groups. this book covers an unusually wide spectrum of topics . the entire presentation is utmost thorough, comprehensive, lucid and absolutely user-friendly. All together, this graduate text his a highly interesting, valuable and welcome addition . (Werner Kleinert, Zentralblatt MATH, Vol. 1053, 2005)

From the reviews of the second edition:

"This is the second, substantially extended edition of the book . The book well deserves to become a standard reference for more researchers working or interested in the theory of abelian varieties." (Fumio Hazama, Mathematical Reviews, 2005c)

"The book under review is the second, essentially augmented edition of the original standardtext, which now also reflects some of the very recent developments. the bibliography has been accordingly up-dated and enhanced. Summing up, the second, amply enlarged and up-dated edition of this outstanding standard monograph on complex abelian varieties has increased its utility in a significant degree, and its leading position among the existing books on the subject has been evidently strengthened." (Werner Kleinert, Zentralblatt MATH, Vol. 1056, 2005)

Zusammenfassung

This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.

Inhalt
Notation.- 1. Complex Tori.- 2. Line Bundles on Complex Tori.- 3. Cohomology of Line Bundles.- 4. Abelian Varieties.- 5. Endomorphisms of Abelian Varieties.- 6. Theta and Heisenberg Groups.- 7. Equations for Abelian Varieties.- 8. Moduli.- 9. Moduli Spaces of Abelian Varieties with Endomorphism Structure.- 10. Abelian Surfaces.- 11. Jacobian Varieties.- 12. Prym Varieties.- 13. Automorphisms.- 14. Vector bundles on Abelian Varieties.- 15. Further Results on Line Bundles an the Theta Divisor.- 16. Cycles on Abelian varieties.- 17. The Hodge Conjecture for General Abelian and Jacobian Varieties.- A. Algebraic Varieties and Complex Analytic Spaces.- B. Line Bundles and Factors of Automorphy.- C. Some Algebraic Geometric Results.- C.1 Some Properties of ?-Divisors.- C.2 The Kodaira Dimension.- C.3 Vanishing Theorems.- C.6 Some Results from Intersection Theory.- C.7 Adjoint Ideals.- D. Derived Categories.- D.1 Definition and First Properties.- D.2 Derived Functors.- D.3 The Grothendieck-Riemann-Roch Theorem.- E. Moduli Spaces of Sheaves.- F. Abelian Schemes.- F.1 Abelian Schemes and the Poincar´e Bundle.- F.2 Relative Fourier Functor.- F.3 The Relative Jacobian.- Glossary of Notation.