Abelian Varieties over the Complex Numbers: A Graduate Course

This textbook offers an introduction to abelian varieties, emphasizing geometric constructions over the complex numbers. It covers complex tori, line bundles, moduli space, examples, the Fourier–Mukai transform, and applications to sheaves and Chow groups. Suitable for first and second graduate courses in algebraic geometry and related subjects.

Format: Paperback / softback
Length: 384 pages
Publication date: 16 March 2023
Publisher: Springer International Publishing AG

This comprehensive textbook delves into the realm of abelian varieties, a fundamental concept in algebraic geometry with immense significance. It emphasizes the construction of important classes of abelian varieties and their algebraic cycles, employing geometric techniques primarily over the complex numbers. The book begins by exploring complex tori and their line bundles, naturally progressing to the definition of abelian varieties. After establishing fundamental properties, the moduli space of abelian varieties is introduced and thoroughly studied. The subsequent chapters are dedicated to examining prominent examples of abelian varieties, including Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Furthermore, the Fourier–Mukai transform is introduced and applied to the study of sheaves, leading to significant results on Chow groups and the Hodge conjecture.

Designed as a primary text for a first course on abelian varieties, this book also serves as an excellent resource for second-year graduate courses in algebraic geometry. Its diverse range of topics and numerous exercises make it suitable for both reading courses and self-study. Moreover, the book offers accessibility to students specializing in algebraic geometry as well as related fields such as number theory, cryptography, mathematical physics, and integrable systems. With its clear explanations, comprehensive coverage, and extensive exercises, this textbook provides a valuable reference for scholars and enthusiasts alike.

Weight: 610g
Dimension: 153 x 235 x 26 (mm)
ISBN-13: 9783031255694
Edition number: 1st ed. 2023