Abelian Varieties over the Complex Numbers

Beschreibung

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 Fourier–Mukai 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.
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 Fourier–Mukai 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 Fourier–Mukai transform for sheaves and applies it to algebraic cycles and the Hodge conjecture Includes more than 300 exercises of varying difficulty

Autor*in

Herbert Lange

Themen in »Abelian Varieties over the Complex Numbers«

textbook on abelian varieties complex abelian varieties Fourier-Mukai transform complex tori Jacobian variety Albanese variety Picard variety Prym variety intermediate Jacobian Hodge conjecture line bundles

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“This is a ‘graduate text’ edition of the book Complex abelian varieties by C. Birkenhake and H. Lange, a major reference in the theory of abelian varieties … . This new edition does offer substantially more exercises …” (Jean Kieffer, Mathematical Reviews, October, 2024)

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