Abelian Varieties over the Complex Numbers

This textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles.
The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the FourierMukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained.
This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.

Presents geometric constructions of the key abelian varieties: Jacobian, Albanese, Picard and Prym varieties Introduces the FourierMukai transform for sheaves and applies it to algebraic cycles and the Hodge conjecture Includes more than 300 exercises of varying difficulty

Auteur
Herbert Lange, now retired, was a professor at the department of mathematics of the University Erlangen-Nuremberg. He has published more than 130 research papers on abelian varieties, curve theory and vector bundles on algebraic curves, as well as several books.

Résumé

"This book is addressed to a wide readership of mathematicians, physicists, students pursuing graduate, masters and higher degrees in mathematics and mathematical physics. It is devoted to some of the theory of abelian varieties. It offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry." (Ahmed Lesfari, ZbMATH 1528.14001, 2024)

"The reorganization of the topics is fine surgical work. Several portions of the original monograph are sewn in a natural way in the new book, adding examples or additional text when necessary, and re-arranging the focus to make it a more friendly introduction to the subject. Careful attention to details and the required background makes the book under review accessible to an interested reader and could be a used as textbook for a course on abelian varieties." (Felipe Zaldivar, MAA Reviews, June 18, 2023)

Contenu

1. Line Bundles on Complex Tori.- 2 Abelian Varieties.- 3 Moduli Spaces.- 4 Jacobian Varieties.- 5 Main Examples of Abelian Varieties.- 6 The Fourier Transform for Sheaves and Cycles.- 7 Introduction to the Hodge Conjecture for Abelian Varieties.